Pith. sign in

REVIEW 3 major objections 5 minor 90 references

Information Propagation and Encoding in Solids: A Quantitative Approach Towards Mechanical Intelligence

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read An elastic solid can be treated as an information encoder: the applied load is the input, the reaction forces at discrete sensors are the output, and normalized mutual information (NMI) measures how much of the load's information reaches th

desk verdict A useful framework for quantifying load-to-sensor information in elastic solids, but the NMI normalization is built from joint X–Y statistics, so the quantitative numbers are provisional until the entropy estimator is corrected. read the letter →

arxiv 2602.00140 v3 pith:TTQNQEIC submitted 2026-01-28 cs.IT cond-mat.mtrl-scimath.IT

Peerasait Prachaseree , Emma Lejeune This is my paper · ORCID
classification cs.ITcond-mat.mtrl-scimath.IT
keywords informationtheorymutualmechanicalintelligenceelasticsolidssensorplacementarchitectedmaterialsrate-distortion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes that an elastic solid, together with discrete force sensors, forms an information channel: the applied load is the input signal and the sensor readings are the output. Because the sensors are assumed noiseless, the mutual information between load and sensors equals the entropy of the sensor readings, and normalizing by the entropy of the load defines a quantity in [0,1] called normalized mutual information (NMI). The central claims are that greedily placed sensors—one per load coefficient—recover essentially all of the load's information (NMI≈1), that information decays with depth in a way consistent with the classical effect that statically equivalent loads become indistinguishable far from the load, and that architected geometry can tune NMI from roughly 0.2 to 0.7, letting structures either transmit or block information. If correct, this gives a task-agnostic, comparable metric for mechanical information processing and a design target for structures that process information through their own deformation.

What carries the argument

The central object is the mechanical information channel and the normalized mutual information NMI=I(X;Y)/h(X). The input is a random load built from coefficients of an orthogonal polynomial basis; the encoder is the elastic body; the output is reaction forces at discrete sensors. With noiseless sensors, I(X;Y)=h(Y), so NMI lies in [0,1] and measures how close the body comes to transmitting the full load information. The greedy sensor-selection algorithm adds sensors one at a time by maximizing conditional mutual information, and the rule that the number of sensors equals the number of load coefficients emerges as the point where NMI saturates. Principal-stress lines act as the geometric cor

What would settle it

Recompute the halfspace NMI with a different estimator (e.g., a histogram or kernel method) on a known ground-truth load distribution and dense sensor sampling, and check whether I(X;Y) stays flat and NMI stays at 1 when the number of sensors exceeds the number of load coefficients; if h(X) drifts with sensor count or NMI drops, the saturation claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that information propagation in an elastic solid has a precise information-theoretic signature. Treating the applied traction (parameterized by polynomial coefficients) as a random source X and the reaction forces at k discrete sensors as the output Y, the mechanical encoder satisfies I(X;Y)=h(Y) because the sensors are noiseless; the theoretical maximum is h(X). The paper defines NMI=I(X;Y)/h(X) and shows numerically that sequentially selecting sensors by conditional mutual information reaches NMI≈1 when k equals the number of load coefficients, so the body transmits the full information content of the load. It further shows that in a halfspace the

Load-bearing premise

The whole normalized measure rests on the assumption that the load's entropy can be estimated in a way that does not depend on the sensor setup; the estimator used here builds its normalization from joint load-sensor statistics, so this assumption is only exact when the load and sensor readings are independent, and it becomes questionable exactly in the high-correlation regime where the paper claims the theoretical maximum.

Editorial extensions

If this is right

  • If the framework is right, the information content of a mechanical load can be measured rather than assumed, and any elastic body can be benchmarked by a single number NMI.
  • Greedy sensor selection with as many sensors as load coefficients is sufficient to capture the full load information in the tested systems, so sensor placement can be guided by conditional mutual information.
  • Architected geometry can act as a tunable information filter: pore-like patterns block information (NMI≈0.2) while slit-like patterns transmit it (NMI≈0.7), giving a mechanical design route for information control.
  • The correlation between principal-stress lines and NMI suggests that load-path visualizations can guide designs for information-transmitting or information-blocking structures.
  • Rate-distortion results indicate that a mechanical encoder plus a learned decoder approaches the theoretical lower bound when sensors are chosen greedily, meaning the body behaves as a near-optimal lossy compressor of load information.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same NMI-style measure could be adapted to quantify mechanical memory by treating time-delayed states as the output, turning the framework from a sensing metric into a general mechanical-information metric.
  • Editorial inference: because the entropy estimator's normalization is built from joint load-sensor statistics, absolute NMI values may shift with sensor count and geometry; comparisons are safest within a fixed setup unless the normalization is independently validated.
  • Editorial inference: the load-path correlation suggests a concrete design rule—route principal-stress lines through sensors for high transmission—which could be tested by deliberately placing a void barrier between load and sensors and measuring the NMI drop.
  • Editorial inference: the fact that optimized structures do not beat simple slits suggests the design parameterization or optimizer, not the physics, is the current bottleneck; more expressive design spaces might approach the theoretical limits more closely.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes an information-theoretic framework for mechanical information transmission: an elastic body is treated as an encoder that maps an applied traction X (parameterized by Legendre coefficients) to noiseless sensor reaction forces Y, and transmission efficiency is quantified by the normalized mutual information NMI = I(X;Y)/h(X). In an elastic halfspace, the authors connect single-sensor mutual information maps to Saint-Venant's principle, propose a greedy conditional-mutual-information sensor selection algorithm, report that k=dx sensors reach the 'theoretical maximum' NMI=1, and complete the channel with a neural-network decoder evaluated with rate-distortion theory. They then use NMI to compare architected geometries (pores and slits) and to drive Bayesian optimization of elliptical voids, reporting normalized mutual information values from about 0.197 to 0.709 (Table 1). The paper includes analytical Flamant-based stress fields, FEniCSx finite-element simulations, a mesh refinement study, and an open-source repository with pytest tests.

Significance. If the normalization were sound, this would be a useful task-agnostic benchmark for mechanical intelligence, with appealing connections to classical mechanics (Saint-Venant's effect, principal stress lines) and to design optimization. Strengths include the clean noiseless-channel identity I(X;Y)=h(Y), the reproducible implementation with publicly available code and tests, the analytical halfspace derivation, mesh refinement checks, and the authors' explicit acknowledgement of the restrictive condition behind their entropy estimator. However, the central normalization relies on a joint-space invariant measure and is therefore configuration-dependent; the quantitative claims and geometry comparisons need substantial rework before they can be taken at face value.

major comments (3)
  1. [§2.2, Eq. (4); Appendix A.3, Eqs. (33)–(37); Appendix C, Fig. 6] The normalization h(X) in NMI=I(X;Y)/h(X) is not an input-intrinsic entropy. The marginal invariant measure m(x) is constructed from joint-space k-nn statistics, specifically from ⟨ρ^{dx+dy}⟩ in Eq. (36), so h_est(X) changes with the sensor set Y, the number of sensors, and the domain geometry. The paper itself states in Appendix C (Fig. 6, right) that for k>dx, I(X;Y) plateaus while h(X) continues to decrease, so NMI=I/h(X) rises because the denominator shrinks. Moreover, the factorization m(x,y)=m(x)m(y) (Eq. 34) is admitted in Appendix A.3 to hold only when X and Y are independent, whereas the mechanical encoder is deterministic and strongly dependent. Thus the claim that k=dx sensors reach the theoretical maximum I(X;Y)/h(X)=1 (Sections 3.1.1, 3.2, 3.3) is at least partly an estimator artifact. Please recompute with a separate estimator of h(X) using only X samples, or use an analyti
  2. [§3.2, Fig. 4; Table 1; §3.3] The cross-geometry comparisons are expressed as NMI=I/h_est(X), where h_est(X) is recomputed for each geometry using joint-space k-nn statistics that depend on Y. The reported spread (e.g., 0.709 vs. 0.197 in Table 1) therefore conflates genuine changes in mutual information with changes in the bias of the denominator. A fair quantitative comparison should report raw I(X;Y) alongside a separately estimated h(X), or use a normalization that is independent of the sensor/output distribution. Without this, the architected-material tuning claim and the Bayesian optimization objectives in Section 3.3 are not quantitatively supported as stated.
  3. [§3.1.1 and Appendix C] The saturation of raw I(X;Y) for k≥dx (Appendix C, left panel) is a useful and defensible numerical observation: additional greedily selected sensors do not increase the mutual information between load and sensor readings. However, this saturation does not by itself establish I(X;Y)=h(X); equality requires h(X) to be the true differential entropy of the load, which is exactly the quantity affected by the invariant-measure issue described above. Please either reframe the k=dx result as saturation of the raw mutual information, or provide an independent estimate of h(X) before claiming that the 'full information content' of the load has been transmitted.
minor comments (5)
  1. [Figure 2 caption vs. §3.1.1] The Figure 2 caption says dx=3 and dx=6, while the text says dx=3 and dx=5 (and Section 3.1.2 uses dx=4). Please make the values consistent.
  2. [Notation conflict] The symbol k denotes the number of sensors in Algorithm 1 and the main text, but denotes the KSG nearest-neighbor count in Appendix A.3. This is confusing because Section 3.1.1 compares k with dx while Appendix A.3 sets k=5. Please use distinct symbols (e.g., n_sensors and k_NN).
  3. [Appendix B.3] Typo: 'in tehe ˆe2 direction' should be 'in the ˆe2 direction'.
  4. [§2.2.2 and elsewhere] 'rate-disortion' should be 'rate-distortion'; 'principle stress lines' should be 'principal stress lines' in several places.
  5. [Appendix H] The noise-variance estimate O(1/N) is described as formally derived for the KSG estimator with the L2 distance metric, while the implementation uses L∞. Please clarify whether this mismatch affects the Bayesian optimization noise model.

Circularity Check

1 steps flagged · score 6.0 of 10

NMI saturation claim is partly an estimator artifact: h(X) is computed from joint X–Y k-nn statistics, so the denominator shrinks with sensor count and inflates I/h(X) toward 1.

  1. self definitional [Section 2.2 Eq. (4); Appendix A.3 Eqs. (34)-(36); Appendix C, Fig. 6]
    "m(x, y) =m(x)m(y)(34) ... m(x) =c −1 dx ( N/k ⟨ρdx+dy ⟩ ) − dx/dx+dy (36) ... However, when k > dx, (maxI(X,Y)/h(X) still increases, but at a slower rate than when k < dx. Note that from the left plot, since I(X,Y) does not change as in the case where k=dx, the I(X,Y)/h(X) comes from the decrease h(X) due to high correlations in between X and Y."

    Eq. (4) defines NMI=I(X;Y)/h(X) and treats h(X) as the theoretical maximum for the mechanical encoder. But the h(X) actually used is the LDDP estimate from Eq. (33)/(37), into which Eq. (36)'s marginal invariant measure m(x) enters. That m(x) is constructed from the joint-space k-nn distance rho in Z=(X,Y), so the 'normalizing' entropy is not an input-intrinsic quantity: it depends on the sensor configuration, the number of sensors k, and the geometry through Y. The paper's own Appendix C confirms that for k>dx, I(X;Y) is flat while h(X) continues to decrease, so the rise of I/h(X) is driven by the denominator. Thus the reported 'approach to the theoretical maximum' and the geometry rankings in Sections 3.2-3.3 are partly an artifact of estimating the normalizer from the same sensor data u

full rationale

The derivation chain is mostly independent: the halfspace solution, Flamant superposition, rate-distortion lower bound, and load-reconstruction benchmark are external checks, and the paper's self-citations ([42], [49], [62], [88]) are background references, not load-bearing. No uniqueness theorem or ansatz is smuggled in via self-citation. The significant issue is the normalized mutual information definition itself. NMI=I(X;Y)/h(X) is presented as a quantity in [0,1] whose upper bound is the information content of the load X. However, the paper's estimator for h(X) uses an invariant measure m(x) derived from joint-space k-nn statistics in (X,Y), so the estimated denominator depends on the sensor outputs and on the number of sensors. Appendix C shows this explicitly: when k>dx, I(X;Y) plateaus but h(X) keeps decreasing, so I/h(X) rises because the denominator shrinks. The paper acknowledges Eq. (34) is restrictive and that the procedure underestimates entropy for highly correlated variables, but it still interprets NMI near 1 as reaching the theoretical maximum and uses it to rank geometries. That part of the central claim is therefore partly circular/estimator-driven. The independent reconstruction results (near-zero MSE for greedy k>=dx) provide non-circular support for the sensor-selection conclusion, which prevents the score from being higher, but the headline 'full information content of the load' claim is not supported by the normalized metric as implemented.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central contribution is a measurement framework; its outputs depend on estimator choices (k, invariant measure) and input-distribution choices more than on physical parameters. The critical load-bearing assumption is the factorized invariant measure, which the authors themselves flag as false for correlated variables.

free parameters (5)
  • KSG nearest-neighbor count k = 5
    Chosen as a balance between bias and variance for the KSG estimator (Appendix A.3); affects all MI/NMI values.
  • Invariant measure m(x) estimated from joint space = data-dependent (Eq. 36)
    The LDDP entropy normalization uses m(x) derived from joint (X,Y) k-nn volumes, making h(X) depend on the sensor configuration; this is the key estimator degree of freedom.
  • Load coefficient distribution = c_n ~ U(-10,10), F=1, a=100, dx=3,4,5,6
    The input source distribution sets h(X) and the absolute MI scale; uniform bounds chosen by the authors, not derived.
  • Bayesian optimization sample count N = 500
    MI is estimated from 500 FE samples per iteration and treated as noisy; the reported convergence and final values inherit this estimator noise (Section 3.3, Appendix H).
  • StandardScaler threshold = 1e-9 magnitude clipped to 0
    Numerical threshold applied to scaled data before KSG estimation (Appendix A.3).
assumptions (4)
  • standard math Linear isotropic elasticity and Flamant superposition describe the half-space response
    Used throughout Section 3.1 and Appendix B; stress is linear in loads, so the encoder map is linear.
  • ad hoc to paper KSG estimator with the factorized invariant measure m(x,y)=m(x)m(y) yields valid entropy estimates
    Eq. 34 in Appendix A.3; the authors explicitly state this holds only when X and Y are independent and that it induces entropy underestimation for correlated variables—the regime central to this study.
  • domain assumption Sensors are noiseless and measure sigma_22 at points at floating-point precision
    Section 2.2: this makes h(Y|X)=0 and I(X;Y)=h(Y); real sensors have noise that would change the information quantities.
  • domain assumption Uniform load expectation allows principal-stress-line visualization under uniform compression to represent expected load paths
    Appendix G: E[X]=0 so uniform compression gives the mean field; the link between principal stress lines and MI is qualitative, not proven.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Information Propagation and Encoding in Solids: A Quantitative Approach Towards Mechanical Intelligence." pith.science (2026). https://pith.science/paper/TTQNQEIC

@misc{pith2026260200140,
  author       = {Pith},
  title        = {Pith review of: Information Propagation and Encoding in Solids: A Quantitative Approach Towards Mechanical Intelligence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTQNQEIC}},
  note         = {Machine review of arXiv:2602.00140}
}
read the original abstract

Engineered systems typically separate mechanical function from information processing, whereas biological systems can exploit physical structure as a medium for information processing and computation. Motivated by this contrast, recent work in mechanics has explored embedding information-processing capabilities directly into mechanical structures. However, quantitative frameworks for evaluating such capabilities remain limited. Here we address a foundational question: how does information propagate through a solid body? Using elastic bodies as a model system, we apply information-theoretic tools to treat an elastic domain as an information encoder and quantify how information transmits from applied loads to discrete sensor locations. We further connect these measures to familiar mechanical phenomena, including Saint-Venant's effect and principal stress lines. Moving toward design, we show how geometry and architected materials can tune transmission, enabling elastic domains to either transmit or block information. Overall, this work advances quantifiable metrics and benchmark tasks for mechanical intelligence, supporting comparable designs of mechanically embodied information processing.

Figures

Figures reproduced from arXiv: 2602.00140 by the authors.

Figure 1
Figure 1. Interpretation of a mechanical system as an information channel. a) Block diagram of a simple information [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Elastic halfspace as an information encoder. a) Schematic of our pipeline to compute mutual information [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Elastic halfspace as an information encoder for a full information channel for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Effect of domain geometry on information transmission. The left panel visualizes a schematic of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Bayesian optimization to maximize and minimize information transmission. a) Left panel visualizes design [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Numerical validation of information bounds using results from the elastic halfspace problem. Left panel [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Results from our mesh refinement study. a) Mesh refinement study of pore geometry. Left panel plots [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Pipeline to compute the principal stress lines used in this work. The major ( [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Raw data from Bayesian optimization. The black line plots data from the maximization optimizer, while the [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 6 linked inside Pith

  1. [1]

    Physics-informed information field theory for modeling physical systems with uncertainty quantification.Journal of Computational Physics, 486:112100, 2023

    Alex Alberts and Ilias Bilionis. Physics-informed information field theory for modeling physical systems with uncertainty quantification.Journal of Computational Physics, 486:112100, 2023. 28 Encoding Information with Mechanical SystemA PREPRINT

    2023

  2. [2]

    Alnaes, Anders Logg, Kristian B

    Martin S. Alnaes, Anders Logg, Kristian B. Ølgaard, Marie E. Rognes, and Garth N. Wells. Unified form language: A domain-specific language for weak formulations of partial differential equations.ACM Transactions on Mathematical Software, 40, 2014

    2014

  3. [3]

    Roadmap on embodying mechano-intelligence and computing in functional materials and structures.Smart Materials and Structures, 34(6):063501, 2025

    Andrea Alù, Andres F Arrieta, Emanuela Del Dottore, Michael Dickey, Samuele Ferracin, Ryan Harne, Helmut Hauser, Qiguang He, Jonathan B Hopkins, Lance P Hyatt, et al. Roadmap on embodying mechano-intelligence and computing in functional materials and structures.Smart Materials and Structures, 34(6):063501, 2025

    2025

  4. [4]

    Perspectives on biological growth and remodeling.Journal of the Mechanics and Physics of Solids, 59(4):863–883, 2011

    D Ambrosi, Gerard A Ateshian, Ellen M Arruda, SC Cowin, J Dumais, A Goriely, Gerhard A Holzapfel, Jay D Humphrey, R Kemkemer, Ellen Kuhl, et al. Perspectives on biological growth and remodeling.Journal of the Mechanics and Physics of Solids, 59(4):863–883, 2011

    2011

  5. [5]

    An algorithm for computing the capacity of arbitrary discrete memoryless channels.IEEE Transactions on Information Theory, 18(1):14–20, 1972

    Suguru Arimoto. An algorithm for computing the capacity of arbitrary discrete memoryless channels.IEEE Transactions on Information Theory, 18(1):14–20, 1972

    1972

  6. [6]

    Petsc users manual

    Satish Balay, Shrirang Abhyankar, Mark Adams, Jed Brown, Peter Brune, Kris Buschelman, Lisandro Dalcin, Alp Dener, Victor Eijkhout, William Gropp, et al. Petsc users manual. 2019

    2019

  7. [7]

    Dolfinx: the next generation fenics problem solving environment

    Igor A Baratta, Joseph P Dean, Jørgen S Dokken, Michal Habera, Jack Hale, Chris N Richardson, Marie E Rognes, Matthew W Scroggs, Nathan Sime, and Garth N Wells. Dolfinx: the next generation fenics problem solving environment. 2023

    2023

  8. [8]

    Mutual information neural estimation

    Mohamed Ishmael Belghazi, Aristide Baratin, Sai Rajeshwar, Sherjil Ozair, Yoshua Bengio, Aaron Courville, and Devon Hjelm. Mutual information neural estimation. InInternational conference on machine learning, pages 531–540. PMLR, 2018

    2018

Show all 90 references
  1. [9]

    Materializing autonomy in soft robots across scales

    Thomas A Berrueta, Todd D Murphey, and Ryan L Truby. Materializing autonomy in soft robots across scales. Advanced Intelligent Systems, 6(2):2300111, 2024

    2024

  2. [10]

    Physical reservoir computing with origami and its application to robotic crawling

    Priyanka Bhovad and Suyi Li. Physical reservoir computing with origami and its application to robotic crawling. Scientific Reports, 11(1):13002, 2021

    2021

  3. [11]

    Computation of channel capacity and rate-distortion functions.IEEE transactions on Information Theory, 18(4):460–473, 2003

    Richard Blahut. Computation of channel capacity and rate-distortion functions.IEEE transactions on Information Theory, 18(4):460–473, 2003

    2003

  4. [12]

    CRC press, 2009

    Allan F Bower.Applied mechanics of solids. CRC press, 2009

    2009

  5. [13]

    Eric Brochu, Vlad M Cora, and Nando De Freitas. A tutorial on bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning.arXiv preprint arXiv:1012.2599, 2010

    2010 arXiv

  6. [14]

    Christopher J Cellucci, Alfonso M Albano, and Paul E Rapp. Statistical validation of mutual information calculations: Comparison of alternative numerical algorithms.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 71(6):066208, 2005

    2005

  7. [15]

    Autonomous deployment of a solar panel using elastic origami and distributed shape-memory-polymer actuators.Physical Review Applied, 11(6):064069, 2019

    Tian Chen, Osama R Bilal, Robert Lang, Chiara Daraio, and Kristina Shea. Autonomous deployment of a solar panel using elastic origami and distributed shape-memory-polymer actuators.Physical Review Applied, 11(6):064069, 2019

    2019

  8. [16]

    John Wiley & Sons, 1999

    Thomas M Cover.Elements of information theory. John Wiley & Sons, 1999

    1999

  9. [17]

    Beyond normal: On the evaluation of mutual information estimators.Advances in neural information processing systems, 36:16957–16990, 2023

    Paweł Czy˙z, Frederic Grabowski, Julia V ogt, Niko Beerenwinkel, and Alexander Marx. Beyond normal: On the evaluation of mutual information estimators.Advances in neural information processing systems, 36:16957–16990, 2023

    2023

  10. [18]

    Extracellular matrix alignment directs provisional matrix assembly and three dimensional fibrous tissue closure

    Shoshana L Das, Prasenjit Bose, Emma Lejeune, Daniel H Reich, Christopher Chen, and Jeroen Eyckmans. Extracellular matrix alignment directs provisional matrix assembly and three dimensional fibrous tissue closure. Tissue Engineering Part A, 27(23-24):1447–1457, 2021

    2021

  11. [19]

    Estimating mutual information using b-spline functions–an improved similarity measure for analysing gene expression data.BMC bioinformatics, 5(1):118, 2004

    Carsten O Daub, Ralf Steuer, Joachim Selbig, and Sebastian Kloska. Estimating mutual information using b-spline functions–an improved similarity measure for analysing gene expression data.BMC bioinformatics, 5(1):118, 2004

    2004

  12. [20]

    Samuel J DePalma, Javiera Jilberto, Austin E Stis, Darcy D Huang, Jason Lo, Christopher D Davidson, Aamilah Chowdhury, Robert N Kent III, Maggie E Jewett, Hiba Kobeissi, et al. Matrix architecture and mechanics regulate myofibril organization, costamere assembly, and contracti...

    2024

  13. [21]

    Information theory for fields.Annalen der physik, 531(3):1800127, 2019

    Torsten A Enßlin. Information theory for fields.Annalen der physik, 531(3):1800127, 2019

    2019

  14. [22]

    A tutorial on bayesian optimization.arXiv preprint arXiv:1807.02811, 2018

    Peter I Frazier. A tutorial on bayesian optimization.arXiv preprint arXiv:1807.02811, 2018. 29 Encoding Information with Mechanical SystemA PREPRINT

    2018 arXiv

  15. [23]

    Efficient estimation of mutual information for strongly dependent variables

    Shuyang Gao, Greg Ver Steeg, and Aram Galstyan. Efficient estimation of mutual information for strongly dependent variables. InArtificial intelligence and statistics, pages 277–286. PMLR, 2015

    2015

  16. [24]

    Bayesian optimization with inequality constraints

    Jacob R Gardner, Matt J Kusner, Zhixiang Eddie Xu, Kilian Q Weinberger, and John P Cunningham. Bayesian optimization with inequality constraints. InICML, volume 2014, pages 937–945, 2014

    2014

  17. [25]

    Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities.International journal for numerical methods in engineering, 79(11):1309–1331, 2009

    Christophe Geuzaine and Jean-François Remacle. Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities.International journal for numerical methods in engineering, 79(11):1309–1331, 2009

    2009

  18. [26]

    Springer Science & Business Media, 1989

    Robert M Gray.Source coding theory, volume 83. Springer Science & Business Media, 1989

    1989

  19. [27]

    Erfan Hamdi and Emma Lejeune. Towards robust surrogate models: Benchmarking machine learning approaches to expediting phase field simulations of brittle fracture.Computer Methods in Applied Mechanics and Engineering, 449:118526, 2026

    2026

  20. [28]

    Towards a theoretical foundation for morphological computation with compliant bodies.Biological cybernetics, 105(5):355–370, 2011

    Helmut Hauser, Auke J Ijspeert, Rudolf M Füchslin, Rolf Pfeifer, and Wolfgang Maass. Towards a theoretical foundation for morphological computation with compliant bodies.Biological cybernetics, 105(5):355–370, 2011

    2011

  21. [29]

    Courier Corporation, 2003

    Thomas JR Hughes.The finite element method: linear static and dynamic finite element analysis. Courier Corporation, 2003

    2003

  22. [30]

    Mechanical cell–cell communication in fibrous networks: the importance of network geometry.Bulletin of mathematical biology, 79(3):498–524, 2017

    DL Humphries, JA Grogan, and EA Gaffney. Mechanical cell–cell communication in fibrous networks: the importance of network geometry.Bulletin of mathematical biology, 79(3):498–524, 2017

    2017

  23. [31]

    Mechanics informatics: A paradigm for efficiently learning constitutive models.Journal of the Mechanics and Physics of Solids, page 106239, 2025

    Royal C Ihuaenyi, Wei Li, Martin Z Bazant, and Juner Zhu. Mechanics informatics: A paradigm for efficiently learning constitutive models.Journal of the Mechanics and Physics of Solids, page 106239, 2025

    2025

  24. [32]

    Edwin T. Jaynes. Information theory and statistical mechanics. In K. W. Ford, editor,Statistical Physics: 1962 Brandeis Lectures in Theoretical Physics, Volume 3. Benjamin, New York, 1963

    1962

  25. [33]

    Fenitop: a simple fenicsx implementation for 2d and 3d topology optimization supporting parallel computing.Structural and Multidisciplinary Optimization, 67(8):140, 2024

    Yingqi Jia, Chao Wang, and Xiaojia Shelly Zhang. Fenitop: a simple fenicsx implementation for 2d and 3d topology optimization supporting parallel computing.Structural and Multidisciplinary Optimization, 67(8):140, 2024

    2024

  26. [34]

    Mechanical metamaterials and beyond.Nature communications, 14(1):6004, 2023

    Pengcheng Jiao, Jochen Mueller, Jordan R Raney, Xiaoyu Zheng, and Amir H Alavi. Mechanical metamaterials and beyond.Nature communications, 14(1):6004, 2023

    2023

  27. [35]

    The rise of intelligent matter.Nature, 594(7863):345–355, 2021

    Corinna Kaspar, Bart Jan Ravoo, Wilfred G van der Wiel, Seraphine V Wegner, and Wolfram HP Pernice. The rise of intelligent matter.Nature, 594(7863):345–355, 2021

    2021

  28. [36]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization.CoRR, abs/1412.6980, 2014

    2014 arXiv

  29. [37]

    Spectral entropy and strain energy trends in composite mechanical metamate- rials.Extreme Mechanics Letters, 45:101289, 2021

    John T Klein and Eduard G Karpov. Spectral entropy and strain energy trends in composite mechanical metamate- rials.Extreme Mechanics Letters, 45:101289, 2021

    2021

  30. [38]

    Estimating mutual information.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 69(6):066138, 2004

    Alexander Kraskov, Harald Stögbauer, and Peter Grassberger. Estimating mutual information.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 69(6):066138, 2004

    2004

  31. [39]

    A structural topology design method based on principal stress line

    Tsz-Ho Kwok, Yongqiang Li, and Yong Chen. A structural topology design method based on principal stress line. Computer-Aided Design, 80:19–31, 2016

    2016

  32. [40]

    Information is physical.Physics Today, 44(5):23–29, 1991

    Rolf Landauer. Information is physical.Physics Today, 44(5):23–29, 1991

    1991

  33. [41]

    Mechanical neural networks: Architected materials that learn behaviors.Science Robotics, 7(71):eabq7278, 2022

    Ryan H Lee, Erwin AB Mulder, and Jonathan B Hopkins. Mechanical neural networks: Architected materials that learn behaviors.Science Robotics, 7(71):eabq7278, 2022

    2022

  34. [42]

    Locality sensitive hashing via mechanical behavior.Extreme Mechanics Letters, 63:102042, 2023

    Emma Lejeune and Peerasait Prachaseree. Locality sensitive hashing via mechanical behavior.Extreme Mechanics Letters, 63:102042, 2023

    2023

  35. [43]

    Dynamic wind-up locomotion enabled by embodied intelligence.The International Journal of Robotics Research, page 02783649251360814, 2025

    Chang Liu and Mark Plecnik. Dynamic wind-up locomotion enabled by embodied intelligence.The International Journal of Robotics Research, page 02783649251360814, 2025

    2025

  36. [44]

    Information-theoretic formulation of dynamical systems: causality, modeling, and control.Physical Review Research, 4(2):023195, 2022

    Adrián Lozano-Durán and Gonzalo Arranz. Information-theoretic formulation of dynamical systems: causality, modeling, and control.Physical Review Research, 4(2):023195, 2022

    2022

  37. [45]

    Force chains in cell–cell mechanical communication.Journal of the Royal Society Interface, 16(159):20190348, 2019

    Amots Mann, Ran S Sopher, Shahar Goren, Ortal Shelah, Oren Tchaicheeyan, and Ayelet Lesman. Force chains in cell–cell mechanical communication.Journal of the Royal Society Interface, 16(159):20190348, 2019

    2019

  38. [46]

    Courier Corporation, 2012

    James Clerk Maxwell.Theory of heat. Courier Corporation, 2012

    2012

  39. [47]

    Bistability-based foldable origami mechanical logic gates.Extreme Mechanics Letters, 43:101180, 2021

    Zhiqiang Meng, Weitong Chen, Tie Mei, Yuchen Lai, Yixiao Li, and CQ Chen. Bistability-based foldable origami mechanical logic gates.Extreme Mechanics Letters, 43:101180, 2021. 30 Encoding Information with Mechanical SystemA PREPRINT

    2021

  40. [48]

    Sympy: symbolic computing in python.PeerJ Computer Science, 3:e103, 2017

    Aaron Meurer, Christopher P Smith, Mateusz Paprocki, Ondˇrej ˇCertík, Sergey B Kirpichev, Matthew Rocklin, AMiT Kumar, Sergiu Ivanov, Jason K Moore, Sartaj Singh, et al. Sympy: symbolic computing in python.PeerJ Computer Science, 3:e103, 2017

    2017

  41. [49]

    Investigating deep learning model calibration for classification problems in mechanics.Mechanics of Materials, 184:104749, 2023

    Saeed Mohammadzadeh, Peerasait Prachaseree, and Emma Lejeune. Investigating deep learning model calibration for classification problems in mechanics.Mechanics of Materials, 184:104749, 2023

    2023

  42. [50]

    The mpmath development team.mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.3.0), 2023.http://mpmath.org/

    2023

  43. [51]

    Accurate estimation of the normalized mutual information of multidimensional data.The Journal of Chemical Physics, 161(5), 2024

    Daniel Nagel, Georg Diez, and Gerhard Stock. Accurate estimation of the normalized mutual information of multidimensional data.The Journal of Chemical Physics, 161(5), 2024

    2024

  44. [52]

    Physical reservoir computing—an introductory perspective.Japanese Journal of Applied Physics, 59(6):060501, 2020

    Kohei Nakajima. Physical reservoir computing—an introductory perspective.Japanese Journal of Applied Physics, 59(6):060501, 2020

    2020

  45. [53]

    Segmenting mechanically heterogeneous domains via unsupervised learning

    Quan Nguyen and Emma Lejeune. Segmenting mechanically heterogeneous domains via unsupervised learning. Biomechanics and Modeling in Mechanobiology, 23(1):349–372, 2024

    2024

  46. [54]

    Bayesian Optimization: Open source constrained global optimization tool for Python, 2014–

    Fernando Nogueira. Bayesian Optimization: Open source constrained global optimization tool for Python, 2014–

    2014

  47. [55]

    Chapman and Hall/CRC, 2017

    J Tinsley Oden and Leszek Demkowicz.Applied functional analysis. Chapman and Hall/CRC, 2017

    2017

  48. [56]

    Neuromechanical autoencoders: Learning to couple elastic and neural network nonlinearity.arXiv preprint arXiv:2302.00032, 2023

    Deniz Oktay, Mehran Mirramezani, Eder Medina, and Ryan P Adams. Neuromechanical autoencoders: Learning to couple elastic and neural network nonlinearity.arXiv preprint arXiv:2302.00032, 2023

    2023 arXiv

  49. [57]

    Compaction through buckling in 2d periodic, soft and porous structures: effect of pore shape

    Johannes Overvelde, Sicong Shan, and Katia Bertoldi. Compaction through buckling in 2d periodic, soft and porous structures: effect of pore shape. 2012

    2012

  50. [58]

    Evolution of acoustic logic gates in granular metamaterials

    Atoosa Parsa, Dong Wang, Corey S O’Hern, Mark D Shattuck, Rebecca Kramer-Bottiglio, and Josh Bongard. Evolution of acoustic logic gates in granular metamaterials. InInternational Conference on the Applications of Evolutionary Computation (Part of EvoStar), pages 93–109. Springer, 2022

    2022

  51. [59]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python.Journal of Machine Learnin...

    2011

  52. [60]

    Morphological computation–connecting brain, body, and environment

    Rolf Pfeifer and Gabriel Gómez. Morphological computation–connecting brain, body, and environment. In Creating brain-like intelligence: From basic principles to complex intelligent systems, pages 66–83. Springer, 2009

    2009

  53. [61]

    Kappa Research, LLC Blue Bell, PA, USA, 2014

    Hossein Pishro-Nik.Introduction to probability, statistics, and random processes. Kappa Research, LLC Blue Bell, PA, USA, 2014

    2014

  54. [62]

    Towards understanding structure-function relationships in random fiber networks.Journal of the Mechanics and Physics of Solids, page 106221, 2025

    Peerasait Prachaseree and Emma Lejeune. Towards understanding structure-function relationships in random fiber networks.Journal of the Mechanics and Physics of Solids, page 106221, 2025

    2025

  55. [63]

    Academic Press, 2009

    Martin H Sadd.Elasticity: theory, applications, and numerics. Academic Press, 2009

    2009

  56. [64]

    Maximizing power and velocity of an information engine.Proceedings of the National Academy of Sciences, 118(20):e2023356118, 2021

    Tushar K Saha, Joseph NE Lucero, Jannik Ehrich, David A Sivak, and John Bechhoefer. Maximizing power and velocity of an information engine.Proceedings of the National Academy of Sciences, 118(20):e2023356118, 2021

    2021

  57. [65]

    Scroggs, Igor A

    Matthew W. Scroggs, Igor A. Baratta, Chris N. Richardson, and Garth N. Wells. Basix: a runtime finite element basis evaluation library.Journal of Open Source Software, 7(73):3982, 2022

    2022

  58. [66]

    Unbiased mechanical cloaks.Proceed- ings of the National Academy of Sciences, 122(19):e2415056122, 2025

    Fernando Vasconcelos Senhora, Emily D Sanders, and Glaucio H Paulino. Unbiased mechanical cloaks.Proceed- ings of the National Academy of Sciences, 122(19):e2415056122, 2025

    2025

  59. [67]

    Chiral nonreciprocal elasticity and mechanical activity.Journal of the Mechanics and Physics of Solids, 171:105163, 2023

    Mohamed Shaat and Harold S Park. Chiral nonreciprocal elasticity and mechanical activity.Journal of the Mechanics and Physics of Solids, 171:105163, 2023

    2023

  60. [68]

    A mathematical theory of communication.The Bell system technical journal, 27(3):379–423, 1948

    Claude E Shannon. A mathematical theory of communication.The Bell system technical journal, 27(3):379–423, 1948

    1948

  61. [69]

    Coding theorems for a discrete source with a fidelity criterion.IRE Nat

    Claude E Shannon et al. Coding theorems for a discrete source with a fidelity criterion.IRE Nat. Conv. Rec, 4(142-163):1, 1959

    1959

  62. [70]

    University of Illinois press, 1998

    Claude E Shannon and Warren Weaver.The mathematical theory of communication. University of Illinois press, 1998

    1998

  63. [71]

    A 99 line topology optimization code written in matlab.Structural and multidisciplinary optimiza- tion, 21(2):120–127, 2001

    Ole Sigmund. A 99 line topology optimization code written in matlab.Structural and multidisciplinary optimiza- tion, 21(2):120–127, 2001

    2001

  64. [72]

    Physical intelligence as a new paradigm.Extreme Mechanics Letters, 46:101340, 2021

    Metin Sitti. Physical intelligence as a new paradigm.Extreme Mechanics Letters, 46:101340, 2021. 31 Encoding Information with Mechanical SystemA PREPRINT

    2021

  65. [73]

    PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK).Journal of Open Source Software, 4(37):1450, May 2019

    Bane Sullivan and Alexander Kaszynski. PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK).Journal of Open Source Software, 4(37):1450, May 2019

    2019

  66. [74]

    Stimuli-responsive materials for biomedical applications

    Adriana Teixeira do Nascimento, Paul R Stoddart, Toon Goris, Miriam Kael, Richard Manasseh, Karen Alt, Jurie Tashkandi, Byung Chul Kim, and Simon E Moulton. Stimuli-responsive materials for biomedical applications. Advanced Materials, 37(36):e07559, 2025

    2025

  67. [75]

    Transition graphs of interacting hysterons: Structure, design, organization and statistics.Royal Society Open Science, 12(9), 2025

    Margot Teunisse and Martin van Hecke. Transition graphs of interacting hysterons: Structure, design, organization and statistics.Royal Society Open Science, 12(9), 2025

    2025

  68. [76]

    The information bottleneck method.arXiv preprint physics/0004057, 2000

    Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method.arXiv preprint physics/0004057, 2000

    2000 arXiv

  69. [77]

    A theory of the learnable.Communications of the ACM, 27(11):1134–1142, 1984

    Leslie G Valiant. A theory of the learnable.Communications of the ACM, 27(11):1134–1142, 1984

    1984

  70. [78]

    Informational active matter.arXiv preprint arXiv:2302.07402, 2023

    Bryan VanSaders, Michel Fruchart, and Vincenzo Vitelli. Informational active matter.arXiv preprint arXiv:2302.07402, 2023

    2023 arXiv

  71. [79]

    Mechanical cloak via data- driven aperiodic metamaterial design.Proceedings of the National Academy of Sciences, 119(13):e2122185119, 2022

    Liwei Wang, Jagannadh Boddapati, Ke Liu, Ping Zhu, Chiara Daraio, and Wei Chen. Mechanical cloak via data- driven aperiodic metamaterial design.Proceedings of the National Academy of Sciences, 119(13):e2122185119, 2022

    2022

  72. [80]

    Bioinspired structural materials

    Ulrike GK Wegst, Hao Bai, Eduardo Saiz, Antoni P Tomsia, and Robert O Ritchie. Bioinspired structural materials. Nature materials, 14(1):23–36, 2015

    2015

  73. [81]

    MIT press Cambridge, MA, 2006

    Christopher KI Williams and Carl Edward Rasmussen.Gaussian processes for machine learning, volume 2. MIT press Cambridge, MA, 2006

    2006

  74. [82]

    Emergent behaviors of buckling-driven elasto-active structures.Proceedings of the National Academy of Sciences, 121(45):e2410654121, 2024

    Yuchen Xi, Tom Marzin, Richard B Huang, Trevor J Jones, and P-T Brun. Emergent behaviors of buckling-driven elasto-active structures.Proceedings of the National Academy of Sciences, 121(45):e2410654121, 2024

    2024

  75. [83]

    Responsive materials architected in space and time

    Xiaoxing Xia, Christopher M Spadaccini, and Julia R Greer. Responsive materials architected in space and time. Nature Reviews Materials, 7(9):683–701, 2022

    2022

  76. [84]

    Principal stress direction- aware streamlined printing path planning for fused filament fabrication.Composites Part A: Applied Science and Manufacturing, 190:108642, 2025

    Guangwen Yan, Jinting Xu, Tong Xiao, Guangwei Zhang, Xin Jiang, and Yuwen Sun. Principal stress direction- aware streamlined printing path planning for fused filament fabrication.Composites Part A: Applied Science and Manufacturing, 190:108642, 2025

    2025

  77. [85]

    Mechanical computing with transmissive snapping of kirigami shells

    Yi Yang, Jin Feng, and Douglas P Holmes. Mechanical computing with transmissive snapping of kirigami shells. Advanced Functional Materials, 34(40):2403622, 2024

    2024

  78. [86]

    Grasping with kirigami shells.Science Robotics, 6(54):eabd6426, 2021

    Yi Yang, Katherine Vella, and Douglas P Holmes. Grasping with kirigami shells.Science Robotics, 6(54):eabd6426, 2021

    2021

  79. [87]

    Mechanical computing.Nature, 598(7879):39–48, 2021

    Hiromi Yasuda, Philip R Buskohl, Andrew Gillman, Todd D Murphey, Susan Stepney, Richard A Vaia, and Jordan R Raney. Mechanical computing.Nature, 598(7879):39–48, 2021

    2021

  80. [88]

    Machine learning-guided design of non-reciprocal and asymmetric elastic chiral metamaterials.International Journal of Solids and Structures, page 113412, 2025

    Lingxiao Yuan, Emma Lejeune, and Harold S Park. Machine learning-guided design of non-reciprocal and asymmetric elastic chiral metamaterials.International Journal of Solids and Structures, page 113412, 2025

    2025

  81. [89]

    Two fundamental mechanisms govern the stiffening of cross-linked networks.Biophysical journal, 108(6):1470–1479, 2015

    Goran Žagar, Patrick R Onck, and Erik Van Der Giessen. Two fundamental mechanisms govern the stiffening of cross-linked networks.Biophysical journal, 108(6):1470–1479, 2015

    2015

  82. [90]

    Physically intelligent autonomous soft robotic maze escaper.Science Advances, 9(36):eadi3254, 2023

    Yao Zhao, Yaoye Hong, Yanbin Li, Fangjie Qi, Haitao Qing, Hao Su, and Jie Yin. Physically intelligent autonomous soft robotic maze escaper.Science Advances, 9(36):eadi3254, 2023. 32

    2023

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.