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The paper claims that embedding a differentiable Fourier heat solver inside a neural surrogate reproduces the mode-resolved phonon Boltzmann transport equation's conductivity map with about 5 percent error from only 300 high-fidelity simula

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Embedding a differentiable Fourier solver as a low-fidelity physics core lets a neural surrogate predict phonon-BTE conductivity of porous nanostructures to ~5% error with 300 BTE simulations and design targets at ~4% average error.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection First credible Fourier-as-low-fidelity surrogate for mode-resolved BTE, with held-out numbers, but reproducibility and coarsening-sensitivity gaps keep me from fully trusting the data-efficiency claim. the 4 major comments →

arxiv 2512.05976 v3 pith:SX775ZN3 submitted 2025-11-25 physics.comp-ph cs.LG

Physics Enhanced Deep Surrogates for the Phonon Boltzmann Transport Equation

Antonio Varagnolo , Giuseppe Romano , Rapha\"el Pestourie This is my paper
classification physics.comp-ph cs.LG
keywords phonon Boltzmann transport equationphysics-enhanced deep surrogatemulti-fidelity surrogate modelingactive learningnanoscale thermal transportinverse designthermal conductivityballistic-diffusive transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper is trying to prove that you do not need to learn nanoscale heat transport from scratch. Its surrogate, PEDS, runs a fast macroscopic Fourier heat solver — accurate to only hundreds of percent in the ballistic regime — and trains a small neural network to correct the solver's input geometry so the pair matches the expensive phonon Boltzmann transport equation. The neural network learns both a geometry transformation and a mixing coefficient that decides how much to trust the Fourier core versus the correction. With uncertainty-driven active learning, the authors report reaching 5% fractional error on test conductivities using 300 high-fidelity simulations, versus thousands for data-only baselines, and they use the surrogate for inverse design of porous silicon over a 12–85 W/mK range with average design errors near 4%. A reader should care because this is a concrete recipe for making repeated PDE-constrained inverse design of nanostructured thermal materials practical, where the bottleneck is the cost of expensive BTE solves.

Core claim

On the paper's own terms, the central discovery is that a deliberately wrong but fast and differentiable physical solver can carry most of the surrogate's burden. The Fourier equation overestimates the effective thermal conductivity of nanoporous silicon by 200% on average and up to almost 700% in the ballistic regime, yet PEDS treats that overestimate as a structured error a small network can undo. The network generates a corrected 5x5 pore topology, mixes it with a coarsened copy of the true geometry through a learned coefficient, and feeds the blend to the Fourier solver; gradients flow through the solver by adjoint differentiation. The result is a surrogate that reaches roughly 5% fracti

What carries the argument

The load-bearing object is the PEDS composition: a differentiable low-fidelity Fourier solver acting on the blended geometry wφ G_NN(p) + (1−wφ) downsample(p), where G_NN is a small fully connected generator, wφ is a learnable mixing coefficient in [0,1], and downsample coarsens the input pore configuration. A deep ensemble with heteroskedastic variance supplies uncertainty estimates that drive active learning. The Fourier solver supplies the inductive bias — boundary conditions and diffusive physics — while the generator supplies the ballistic correction; adjoint differentiation makes the solver trainable end-to-end. The mixing coefficient is more than an architectural detail: its learned v

Load-bearing premise

The load-bearing premise is that the 5x5 coarsened Fourier conductivity field of a geometry carries enough information for the small neural generator to correct Fourier's hundreds-of-percent ballistic errors down to about 5% for all geometries in the class — an assumption the paper motivates but does not test with a resolution sensitivity study.

What would settle it

Re-run the same experiments while varying the Fourier solver's spatial resolution (e.g., 3x3, 5x5, 7x7, 100x100) and count how many BTE-labeled designs are needed to hit 5% error at each resolution; if the 300-point result degrades sharply or requires a much larger generator at finer grids, the coarse-representation assumption is false. A second check would be to generate out-of-distribution geometries with pores smaller than 10 nm or non-square shapes and see whether the 5% error holds.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • At data budgets around 300–1000 BTE solves, PEDS with active learning reaches lower test fractional error than MLP and Gaussian process baselines, and the gap is largest in the small-data regime.
  • Training on only low-conductivity (ballistic) samples yields 5.1% error on mid-range and 12.7% on high-conductivity out-of-distribution test sets, roughly a factor of 4 better than a pure MLP, so the physics core transfers to unseen regimes.
  • Eight inverse-design campaigns for target conductivities between 12 and 85 W/mK complete in minutes rather than hours, with PEDS+AL average design error about 4%, close to the about 2.4% of direct BTE optimization.
  • Training cost is amortized: around four design runs pay for the surrogate's fixed training cost, after which each new design is nearly free.
  • The learned mixing coefficient correlates linearly with the Knudsen number, giving a physical readout of when Fourier's equation stops being valid.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The architecture implies a general recipe: any fast solver that shares boundary structure with a costly PDE can serve as a physics prior, provided its systematic error is smooth enough in the design space for a small generator to correct; the natural next test is whether the same 300-point efficiency holds for continuous, higher-dimensional pore parameterizations or for other materials.
  • The Knudsen-number correlation suggests the mixing coefficient could be frozen or replaced by a physical formula, potentially yielding a parameter-free hybrid that needs even fewer labeled data points — an extension the paper does not test.
  • Since the Fourier core is thousands of times faster and the correction is learned in input space, the surrogate could act as a learned preconditioner or warm start for iterative BTE solvers, converting offline training into online speedups, which the paper mentions as ongoing work.
  • Transfer to electron transport, neutron transport, or rarefied gas dynamics is hypothesized; if a drift-diffusion or continuum core plays the same corrective role, the data-efficiency result may generalize beyond phonons, but that remains the authors' speculation, not a demonstrated result.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces PEDS, a physics-enhanced deep surrogate for the phonon Boltzmann transport equation (BTE). The architecture (Eq. 1) combines a differentiable Fourier solver as a low-fidelity physics core with a neural generator that transforms the design variables, plus a learned mixing coefficient wφ that interpolates between the original and generated geometries. The model is trained end-to-end, equipped with deep-ensemble uncertainty quantification, and used with active learning to reduce the number of high-fidelity BTE simulations. The central claims are that PEDS+AL reaches ~5% fractional error with only 300 BTE evaluations, reduces training-data requirements by up to 70% relative to purely data-driven baselines, and enables inverse design of 25-binary-parameter porous silicon geometries spanning 12–85 W m−1 K−1 with average design errors of 4%. The paper also claims that the learned mixing coefficient recovers the ballistic-to-diffusive transition and that PEDS generalizes better than MLP and GP baselines out of distribution.

Significance. If the claims hold, this is a useful contribution to surrogate-based nano-scale thermal design: it demonstrates that a very cheap Fourier core plus a learned input correction can be substantially more data-efficient than pure operator learning or black-box regression, and it provides a concrete cost-benefit analysis with a break-even of about four design runs. The paper has real strengths: evaluations on held-out test sets, statistics over five seeds, comparisons to GP and MLP baselines with and without active learning, and an open GitHub repository. These make the main predictive claims reproducible in principle, although the Fourier solver used for final training is not yet released.

major comments (4)
  1. [Sec. 4.1, Table 1] The text states that PEDS+AL "reduces the fractional test error by approximately 70% compared to PEDS without active learning and by about 75% compared to a purely data-driven MLP trained with active learning." These numbers do not follow from Table 1. At 300 evaluations, the relative reduction versus PEDS-ENS is about 25% ((6.72−5.05)/6.72), and versus MLP+AL about 47% ((9.62−5.05)/9.62). Similar values hold at the other budgets. The 70–75% figures are plausible only for the reduction in training data needed to reach ~5% error (roughly 1000→300 points), not for fractional test error. Please correct the wording or give the exact basis for the claim.
  2. [Sec. 3.1] The low-fidelity Fourier solver input is a 5×5 coarsened conductivity grid, stated to be "the minimum required to adequately represent the 5×5 pore configuration," but no sensitivity study is presented. The data-efficiency claim rests on this representation containing enough information for the generator to correct Fourier errors that range up to ~700%. Without varying the coarsening resolution (e.g., 3×3, 5×5, 10×10) or the Fourier solver discretization/tolerance, the 300-point result is a single point estimate, and it is unclear whether it generalizes to other geometry classes or solver settings. Please add such a sensitivity analysis or explicitly delimit the scope of the claim.
  3. [Sec. 4.2, Table 2] The statement that the reported design errors are "dominated by the material fabrication error" is not supported by any fabrication-error model or quantitative estimate. Moreover, the design errors for targets 12.0 W/mK (7%) and 85.0 W/mK (10%) exceed the stated 5% target; the 4.0% average conceals a large spread. Please either define the fabrication-error distribution used and compare the surrogate errors to it, or qualify the claim appropriately.
  4. [Sec. 5, Figs. 5b and 6] The interpretability claim that the learned mixing coefficient wφ||G|| "discovers" the ballistic-diffusive transition is weakened by the fact that wφ is trained end-to-end to reproduce κ_BTE. A correlation between a fitted parameter and the Knudsen number is expected to some degree. To support the discovery claim, please show that the correlation persists for models trained without the Fourier core, or test on out-of-sample geometries. As written, this is presented as a consistency check rather than a discovered law.
minor comments (4)
  1. [Abstract and Sec. 1] The conductivity range is given as 12–85 W/mK in the abstract but 12–90 W/mK in the introduction. Table 2 targets end at 85. Please unify.
  2. [Eq. (1)] Typo: "flow f id" should be "f_low_fid"; also "wφ dapts" should be "wφ adapts".
  3. [Sec. 6] Typo: "nerual operator" should be "neural operator"; "the the low-fidelity model" has a duplicated article.
  4. [Sec. 8] The Fourier solver used for final training is not yet released, and data are available only upon request. This limits reproducibility. Consider releasing a reference implementation or a detailed parameter list.

Circularity Check

1 steps flagged

Held-out predictive and design results are genuine; the interpretability claim reduces the fitted mixing coefficient to a Knudsen number constructed from the same training labels and a self-cited relation.

specific steps
  1. fitted input called prediction [Sec. 5 Model Interpretability, Figs. 5b and 6; Eq. (1), Sec. 2.1 NLL loss; Knudsen relation from Ref. [77]]
    "Its nonlinear dependence on κBT E demonstrates that the model has successfully learned the transition between transport regimes: large values of wφ ∥G∥ correspond to low-conductivity (ballistic-dominated) cases... For each structure, the following equality holds: κBT E/κFourier = 1+Kn(lnKn−1)/(Kn−1)^2 ... In Fig. 6, the Knudsen number is computed for a set of representative geometries and shown to correlate strongly with the generated ballistic correction wφ ∥G∥."

    wφ and the generator are trained end-to-end so that Eq. (1) matches the BTE ground-truth κ_BTE through the NLL loss in Sec. 2.1. Kn is not measured independently; it is defined by inverting κ_BTE/κ_Fourier via the relation from Ref. [77], which shares an author with this paper. The reported correlation between the fitted correction wφ∥G∥ and Kn therefore connects a fitted quantity to a quantity built from the same training label κ_BTE and the fixed low-fidelity κ_Fourier. Any surrogate that accurately fits κ_BTE must encode the κ_BTE-versus-κ_Fourier discrepancy, so the claimed recovery of the ballistic-diffusive transition is the fit itself, not an independent physical discovery. The self-cited origin of the Kn formula does not remove this construction.

full rationale

The central data-efficiency and inverse-design claims are evaluated on held-out geometries (Table 1) and on Bayesian-optimization campaigns measured against BTE ground truth (Table 2), so the 300-point 5% error result and the 4% design error are not circular. The 5×5 coarsening-sufficiency statement and the absence of a Fourier-resolution sensitivity study are generalization risks, not circularity. The only substantive circular step is in the interpretability narrative: the mixing coefficient is fitted to κ_BTE, and the Knudsen number with which it is correlated is defined from the same κ_BTE/κ_Fourier pair through a same-group citation. This affects a supporting interpretability claim rather than the surrogate's held-out accuracy, hence a moderate score of 4.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

No new physical entities are introduced. The learned mixing coefficient is a model parameter, not a postulated physical quantity. The key free parameters are the neural-network weights, the mixing coefficient, the NLL weight, the AL hyperparameters, and the Fourier coarsening resolution. The central claim rests on the domain assumption that the coarse Fourier solver, plus a small learned input correction, can represent the BTE conductivity map for the 25-parameter geometry family.

free parameters (5)
  • mixing coefficient wφ(x) = learned, constrained to [0,1]
    Controls the convex combination of the neural-generator output and the downsampled original geometry in Eq. 1. It is fitted end-to-end to training data; the interpretability claim that wφ||G|| recovers the ballistic-diffusive transition depends on this learned parameter.
  • neural generator weights (generatorNN) = learned (two hidden layers of 64 units, ReLU, hard-tanh output)
    Maps the 25 binary geometry parameters to a modified 5×5 topology before entering the Fourier solver. All of these weights are fitted to the BTE-labeled training set.
  • heteroskedastic variance network and NLL weight β = β∈[0,1], value not specified
    The negative log-likelihood loss in Sec. 2.1 uses β to balance variance penalty and residual fit. The active-learning query selection relies on the predicted variance, so this hyperparameter affects the reported data-efficiency numbers.
  • active-learning hyperparameters (initial N, proposal M, selection K) = not specified
    The AL loop is described qualitatively in Sec. 2.2: initialize with N points, propose M new geometries, add the K most uncertain. The 300-point result depends on these unstated choices.
  • Fourier coarsening resolution = 5×5 grid
    The low-fidelity Fourier solver uses a 5×5 conductivity discretization versus the 100×100 BTE grid. The paper asserts this is the minimum resolution needed to represent the 5×5 pore configuration, but no sensitivity study is provided.
axioms (6)
  • domain assumption The mode-resolved relaxation-time-approximation BTE (Eq. 2) with almaBTE first-principles phonon data is an accurate ground-truth model for nanoscale heat transport in silicon.
    All high-fidelity labels are generated with OpenBTE using this model; the surrogate's accuracy is measured against it. The paper does not validate this BTE model against experiment beyond citing established practice.
  • domain assumption The Fourier equation is an appropriate low-fidelity limit of the BTE, and a 5×5 coarsened Fourier solver retains enough physical structure to serve as an inductive bias.
    The entire PEDS design (Eq. 1, Sec. 2) assumes the Fourier solver is a useful base that a small neural correction can fix, despite its average 220% error and up to ~700% error in the ballistic regime.
  • domain assumption The ensemble predictive variance is a good proxy for surrogate error and is suitable for active-learning sample selection.
    Sec. 2.2 selects the K most uncertain geometries using the variance from the deep ensemble. The data-efficiency gains reported in Table 1 depend on this uncertainty estimate being informative.
  • domain assumption The 5×5 binary-pore parameterization with L=100 nm spans the claimed design range and is representative of relevant porous nanostructures.
    The paper designs conductivities from 12 to 85 W/mK using 25 binary parameters (Sec. 3.1). The relevance of this restricted geometry family to real fabrication targets is assumed.
  • domain assumption The relation κ_BTE/κ_Fourier = 1+Kn(lnKn−1)/(Kn−1)^2 from Ref. [77] correctly defines the Knudsen number for mode-resolved systems.
    The interpretability claim in Sec. 5 uses this equality to compute Kn from κ_BTE and κ_Fourier. The result is taken from a paper by the same group and is not re-derived or independently validated here.
  • ad hoc to paper The reported design errors are 'dominated by the material fabrication error.'
    This assertion appears in Sec. 4.2 and the abstract, but no fabrication-error model, measurement, or citation is provided. It is not needed for the core surrogate claim but is used to contextualize the 4% design error.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Physics Enhanced Deep Surrogates for the Phonon Boltzmann Transport Equation." pith.science (2026). https://pith.science/paper/SX775ZN3

@misc{pith2026251205976,
  author       = {Pith},
  title        = {Pith review of: Physics Enhanced Deep Surrogates for the Phonon Boltzmann Transport Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SX775ZN3}},
  note         = {Machine review of arXiv:2512.05976}
}
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abstract

Designing materials with controlled heat flow at the nano-scale is central to advances in microelectronics, thermoelectrics, and energy-conversion technologies. At these scales, phonon transport follows the Boltzmann Transport Equation (BTE), which captures non-diffusive (ballistic) effects but is too costly to solve repeatedly in inverse-design loops. Existing surrogate approaches trade speed for accuracy: fast macroscopic solvers can overestimate conductivities by hundreds of percent, while recent data-driven operator learners often require thousands of high-fidelity simulations. This creates a need for a fast, data-efficient surrogate that remains reliable across ballistic and diffusive regimes. We introduce a Physics-Enhanced Deep Surrogate (PEDS) that combines a differentiable Fourier solver with a neural generator and couples it with uncertainty-driven active learning. The Fourier solver acts as a physical inductive bias, while the network learns geometry-dependent corrections and a mixing coefficient that interpolates between macroscopic and nano-scale behavior. PEDS reduces training-data requirements by up to 70% compared with purely data-driven baselines, achieves roughly 5% fractional error with only 300 high-fidelity BTE simulations, and enables efficient design of porous geometries spanning 12-85 W m$^{-1}$ K$^{-1}$ with average design errors of 4%. The learned mixing parameter recovers the ballistic-diffusive transition and improves out of distribution robustness. These results show that embedding simple, differentiable low-fidelity physics can dramatically increase surrogate data-efficiency and interpretability, making repeated PDE-constrained optimization practical for nano-scale thermal-materials design.

Figures

Figures reproduced from arXiv: 2512.05976 by Antonio Varagnolo, Giuseppe Romano, Rapha\"el Pestourie.

Figure 1
Figure 1. Figure 1: PEDS Diagram The main panel illustrates the PEDS workflow: starting from a vector of topology parameters G, a neural-network generator generatorNN produces an updated, coarse topology that encodes a learned, non-linear transformation of the design space. This generated topology is then combined with a coarsified representation of the original geometry via a linear combination; the mixing coefficient wφ ∈ [… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between the Fourier solver and the BTE. On the left we show the qualitative difference [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Panel (a) quantifies data efficiency by plotting fractional test error against training-set size for three [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Training was performed on geometries from one segment of the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) PCA of the generated conductivities projected on the first two principal components and revealing a [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The generated ballistic correction w∥G∥ is highly linearly correlated with the Knudsen number of the geometry (dashed line). The higher the Knudsen number, the bigger is the MFP compared to the representative feature size and the more important will be the ballistic correction to the diffusive model. This confirms that PEDS is able to recover the transition between diffusive and ballistic transport and com… view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

98 extracted references · 18 canonical work pages · 1 internal anchor

  1. [1]

    Zur kinetischen Theorie der Wärmeleitung in Kristallen

    Rudolf Peierls. “Zur kinetischen Theorie der Wärmeleitung in Kristallen”. In:Annalen der Physik395.8 (1929), pp. 1055–1101.DOI:10.1002/andp.19293950803

    doi:10.1002/andp.19293950803 1929

  2. [2]

    Mitchell.The Need for Biases in Learning Generalizations

    Tom M. Mitchell.The Need for Biases in Learning Generalizations. Tech. rep. Rutgers University Technical Report CBM-TR-117, 1980

    1980

  3. [3]

    Efficient Global Optimization of Expensive Black-Box Functions

    Donald R. Jones, Matthias Schonlau, and William J. Welch. “Efficient Global Optimization of Expensive Black-Box Functions”. In:Journal of Global Optimization. V ol. 13. 4. 1998, pp. 455–492.DOI: 10. 1023/A:1008306431147

    1998

  4. [4]

    Neural space-mapping optimization for EM-based design

    Mohamed H Bakr et al. “Neural space-mapping optimization for EM-based design”. In:IEEE Transac- tions on Microwave Theory and Techniques48.12 (2000), pp. 2307–2315

    2000

  5. [5]

    Predicting the output from a complex computer code when fast approxi- mations are available

    M. Kennedy and A. O’Hagan. “Predicting the output from a complex computer code when fast approxi- mations are available”. In:Biometrika87.1 (2000), pp. 1–13.DOI:10.1093/biomet/87.1.1

    doi:10.1093/biomet/87.1.1 2000

  6. [6]

    J. M. Ziman.Electrons and Phonons: The Theory of Transport Phenomena in Solids. Oxford University Press, 2001.ISBN: 9780198507796

    2001

  7. [7]

    Nanoscale thermal transport

    D. G. Cahill et al. “Nanoscale thermal transport”. In:Journal of Applied Physics93.2 (2003), pp. 793– 818

    2003

  8. [8]

    Thermal conductivity of periodic microporous silicon films

    D. Song and G. Chen. “Thermal conductivity of periodic microporous silicon films”. In:Applied Physics Letters84.5 (2004), pp. 687–689.DOI:10.1063/1.1642753

    work page doi:10.1063/1.1642753 2004

  9. [9]

    Chen.Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons

    G. Chen.Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. Available at https://www.amazon.com/dp/019515942X. Oxford University Press, 2005.ISBN: 9780195159424

    arXiv 2005

  10. [10]

    Lattice Boltzmann simulation of rarefied gas flows in microchannels

    Yonghao Zhang, Rongshan Qin, and David R Emerson. “Lattice Boltzmann simulation of rarefied gas flows in microchannels”. In:Physical Review E—Statistical, Nonlinear, and Soft Matter Physics71.4 (2005), p. 047702

    2005

  11. [11]

    Bishop.Pattern Recognition and Machine Learning

    Christopher M. Bishop.Pattern Recognition and Machine Learning. New York: Springer, 2006.ISBN: 978-0387310732

    2006

  12. [12]

    Topology optimization of heat conduc- tion problems using the finite volume method

    Allan Gersborg-Hansen, Martin P. Bendsøe, and Ole Sigmund. “Topology optimization of heat conduc- tion problems using the finite volume method”. In:Structural and Multidisciplinary Optimization31.4 (2006), pp. 251–259.DOI:10.1007/s00158-005-0584-3

    doi:10.1007/s00158-005-0584-3 2006

  13. [13]

    Statistical Improvement Criteria for Use in Multiobjective Design Optimization

    Andy J. Keane and Alexander I. J. Forrester. “Statistical Improvement Criteria for Use in Multiobjective Design Optimization”. In:Computational Engineering and Design Group, University of Southampton (2006). Extends expected improvement criteria applied to multi-objective design problems using kriging surrogates

    2006

  14. [14]

    Carl Edward Rasmussen and Christopher K. I. Williams.Gaussian Processes for Machine Learning. MIT Press, 2006. 18

    2006

  15. [15]

    Multi-fidelity optimization via surrogate modelling

    A. I. J. Forrester, A. Sóbester, and A. J. Keane. “Multi-fidelity optimization via surrogate modelling”. In: Proceedings of the Royal Society A463 (2007), pp. 3251–3269.DOI:10.1098/rspa.2007.1900

    arXiv 2007

  16. [16]

    Andreas Griewank and Andrea Walther.Evaluating Derivatives: Principles and Techniques of Al- gorithmic Differentiation. 2nd ed. Philadelphia, PA: SIAM, 2008.ISBN: 978-0-89871-659-7.DOI: 10.1137/1.9780898717761

    doi:10.1137/1.9780898717761 2008

  17. [17]

    Enhanced thermoelectric performance of rough silicon nanowires

    A. I. Hochbaum et al. “Enhanced thermoelectric performance of rough silicon nanowires”. In:Nature 451.7175 (2008), pp. 163–167.DOI: 10 . 1038 / nature06381.URL: http : / / www . nature . com / nature/journal/v451/n7175/abs/nature06381.html

    2008

  18. [18]

    Space mapping

    Slawomir Koziel, Qingsha S Cheng, and John W Bandler. “Space mapping”. In:IEEE Microwave Magazine9.6 (2008), pp. 105–122

    2008

  19. [19]

    Topology optimization for nano-scale heat transfer

    A. Evgrafov et al. “Topology optimization for nano-scale heat transfer”. In:International Journal for Numerical Methods in Engineering77.2 (2009), pp. 285–300.DOI:10.1002/nme.2413

    work page doi:10.1002/nme.2413 2009

  20. [20]

    Recent Advances in Surrogate-Based Optimization

    Alexander I. J. Forrester and Andy J. Keane. “Recent Advances in Surrogate-Based Optimization”. In: Progress in Aerospace Sciences45.1-3 (2009), pp. 50–79

    2009

  21. [21]

    Trevor Hastie, Robert Tibshirani, and Jerome Friedman.The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd ed. Springer, 2009.ISBN: 9780387848570

    2009

  22. [22]

    Holey Silicon as an Efficient Thermoelectric Material

    Jinyao Tang et al. “Holey Silicon as an Efficient Thermoelectric Material”. In:Nano Letters10.10 (2010), pp. 4279–4283.DOI:10.1021/nl102931z

    work page doi:10.1021/nl102931z 2010

  23. [23]

    Nanostructured thermoelectrics: big efficiency gains from small features

    C. J. Vineis et al. “Nanostructured thermoelectrics: big efficiency gains from small features”. In: Advanced Materials22 (2010), pp. 3970–3980.DOI:10.1002/adma.201000839

    work page doi:10.1002/adma.201000839 2010

  24. [24]

    Convergence of cell based finite volume discretiza- tions for problems of control in the conduction coefficients

    A. Evgrafov, M. M. Gregersen, and M. P. Sørensen. “Convergence of cell based finite volume discretiza- tions for problems of control in the conduction coefficients”. In:ESAIM: Mathematical Modelling and Numerical Analysis45.6 (2011), pp. 1059–1080.DOI:10.1051/m2an/2011012

    arXiv 2011

  25. [25]

    Synthesis Lectures on Artificial Intelligence and Machine Learning

    Burr Settles.Active Learning. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers, 2012.DOI:10.1007/978-3-031-01560-1

    doi:10.1007/978-3-031-01560-1 2012

  26. [26]

    Practical Bayesian optimization of machine learning algorithms

    Jasper Snoek, Hugo Larochelle, and Ryan P Adams. “Practical Bayesian optimization of machine learning algorithms”. In:Advances in Neural Information Processing Systems. V ol. 25. 2012

    2012

  27. [27]

    Nanoscale thermal transport. II. 2003–2012

    David G Cahill et al. “Nanoscale thermal transport. II. 2003–2012”. In:Applied Physics Reviews1.1 (2014), p. 011305

    2003

  28. [28]

    Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification

    Kaiming He et al. “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification”. In:Proceedings of the IEEE International Conference on Computer Vision (ICCV). 2015, pp. 1026–1034

    2015

  29. [29]

    Ballistic Phonon Transport in Holey Silicon

    J. Lee, J. Lim, and P. Yang. “Ballistic Phonon Transport in Holey Silicon”. In:Nano Letters15.5 (2015), pp. 3273–3279.DOI: 10.1021/acs.nanolett.5b00495 .URL: https://doi.org/10.1021/acs. nanolett.5b00495

    work page doi:10.1021/acs.nanolett.5b00495 2015

  30. [30]

    Group equivariant convolutional networks

    Taco Cohen and Max Welling. “Group equivariant convolutional networks”. In:Proceedings of the 33rd International Conference on Machine Learning (ICML). 2016

    2016

  31. [31]

    Gaussian process regression for machine learning: theory and applications

    Jun Wang et al. “Gaussian process regression for machine learning: theory and applications”. In: Mathematics4.1 (2016), p. 34.DOI:10.3390/math4010034. 19

    work page doi:10.3390/math4010034 2016

  32. [32]

    All-in-one energy harvesting and storage devices

    Zhong Lin Wang, Zong-Hong Wang, and Ya Yang. “All-in-one energy harvesting and storage devices”. In:Journal of Materials Chemistry A4.38 (2016), pp. 14686–14704.DOI:10.1039/C6TA01229A

    work page doi:10.1039/c6ta01229a 2016

  33. [33]

    Geometric deep learning: going beyond Euclidean data

    Michael M. Bronstein et al. “Geometric deep learning: going beyond Euclidean data”. In:IEEE Signal Processing Magazine34.4 (2017), pp. 18–42

    2017

  34. [34]

    almaBTE: A solver of the space–time dependent Boltzmann transport equation for phonons in structured materials

    Jesús Carrete et al. “almaBTE: A solver of the space–time dependent Boltzmann transport equation for phonons in structured materials”. In:Computer Physics Communications220 (2017), pp. 351–362

    2017

  35. [35]

    Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles

    Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. “Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles”. In:Advances in Neural Information Processing Systems. V ol. 30. 2017, pp. 6402–6413.URL: https :// proceedings .neurips .cc /paper /2017 /file/ 9ef2ed4b7fd2c810847ffa5fa85bce38-Paper.pdf

    2017

  36. [36]

    Automatic differentiation in machine learning: a survey

    Atilim Gunes Baydin et al. “Automatic differentiation in machine learning: a survey”. In:Journal of Machine Learning Research18.153 (2018), pp. 1–43.URL: http://jmlr.org/papers/v18/17- 468.html

    2018

  37. [37]

    A density-based topology optimization methodology for thermoelectric energy conversion problems

    Christian Lundgaard and Ole Sigmund. “A density-based topology optimization methodology for thermoelectric energy conversion problems”. In:Structural and Multidisciplinary Optimization57.4 (2018), pp. 1427–1442.DOI:10.1007/s00158-018-1919-1

    work page doi:10.1007/s00158-018-1919-1 2018

  38. [38]

    Nanophotonic particle simulation and inverse design using artificial neural net- works

    John Peurifoy et al. “Nanophotonic particle simulation and inverse design using artificial neural net- works”. In:Science Advances4.6 (2018), eaar4206.DOI:10.1126/sciadv.aar4206

    doi:10.1126/sciadv.aar4206 2018

  39. [39]

    Objective-free design of nanophotonic devices with generative adversarial networks

    Shuxiang An, Bolei Zheng, and Nicholas X. Fang. “Objective-free design of nanophotonic devices with generative adversarial networks”. In:ACS Photonics6.11 (2019), pp. 3196–3207.DOI:10.1021/ acsphotonics.9b00824

    2019

  40. [40]

    Coarse-and fine-mesh space mapping for EM optimization incorporating mesh deformation

    Feng Feng et al. “Coarse-and fine-mesh space mapping for EM optimization incorporating mesh deformation”. In:IEEE Microwave and Wireless Components Letters29.8 (2019), pp. 510–512

    2019

  41. [41]

    Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures

    Yousef Kiarashinejad, Sajjad Abdollahramezani, and Ali Adibi. “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures”. In:ACS Photonics6.12 (2019), pp. 3017–3029.DOI:10.1021/acsphotonics.9b00906

    work page doi:10.1021/acsphotonics.9b00906 2019

  42. [42]

    Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design

    Turab Lookman et al. “Active learning in materials science with emphasis on adaptive sampling using uncertainties for targeted design”. In:npj Computational Materials5.1 (2019), p. 21

    2019

  43. [43]

    Learning nonlinear operators via DeepONet

    Lu Lu et al. “Learning nonlinear operators via DeepONet”. In:arXiv preprint arXiv:1910.03193(2019)

    Pith/arXiv arXiv 1910

  44. [44]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”. In:Journal of Computational Physics378 (2019), pp. 686–707.DOI: 10.1016/j.jcp. 2018.10.045

    doi:10.1016/j.jcp 2019

  45. [45]

    Deep learning model to predict complex stress and strain fields in hierarchical composites

    Shuxiang An et al. “Deep learning model to predict complex stress and strain fields in hierarchical composites”. In:Science Advances6.12 (2020), eaaz2540.DOI:10.1126/sciadv.aaz2540

    work page doi:10.1126/sciadv.aaz2540 2020

  46. [46]

    Deep learning reveals underlying physics of light–matter interactions in nanophotonics

    Yousef Kiarashinejad et al. “Deep learning reveals underlying physics of light–matter interactions in nanophotonics”. In:Advanced Theory and Simulations3.7 (2020), p. 2000173.DOI: 10.1002/adts. 202000173

    doi:10.1002/adts 2020

  47. [47]

    Knowledge discovery in nanophotonics using neural networks

    Yousef Kiarashinejad et al. “Knowledge discovery in nanophotonics using neural networks”. In:ACS Photonics7.8 (2020), pp. 2013–2019.DOI:10.1021/acsphotonics.0c00824. 20

    work page doi:10.1021/acsphotonics.0c00824 2020

  48. [48]

    Fourier Neural Operator for Parametric Partial Differential Equations

    Zongyi Li et al. “Fourier Neural Operator for Parametric Partial Differential Equations”. In:arXiv preprint arXiv:2010.08895(2020)

    Pith/arXiv arXiv 2010

  49. [49]

    Physics Informed Neural Networks for Simulating Radiative Transfer

    Siddhartha Mishra and Roberto Molinaro. “Physics Informed Neural Networks for Simulating Radiative Transfer”. In:arXiv preprint arXiv:2009.13291(2020)

    Pith/arXiv arXiv 2009

  50. [50]

    Active Learning of Deep Surrogates for PDEs: Application to Metasurface Design

    R. Pestourie et al. “Active Learning of Deep Surrogates for PDEs: Application to Metasurface Design”. In:NPJ Computational Materials6.1 (2020).DOI:10.1038/s41524-020-00431-2

    doi:10.1038/s41524-020-00431-2 2020

  51. [51]

    Deep Active Learning: Unified and Principled Method for Query and Training

    Changjian Shui et al. “Deep Active Learning: Unified and Principled Method for Query and Training”. In:Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS). Ed. by Silvia Chiappa and Roberto Calandra. V ol. 108. Proceedings of Machine Learning Research. PMLR, 2020, pp. 1308–1318.URL:https://proceedings.mlr.p...

    2020

  52. [52]

    Deep learning enabled inverse design in nanophotonics

    Sunae So et al. “Deep learning enabled inverse design in nanophotonics”. In:Nanophotonics. V ol. 9. 5. 2020, pp. 1041–1057.DOI:10.1515/nanoph-2019-0474

    doi:10.1515/nanoph-2019-0474 2020

  53. [53]

    Genetic Algorithm–Driven Discovery of Unexpected Thermal Conductivity Enhancement by Disorder in Nanoporous Graphene

    Han Wei, Hua Bao, and Xiulin Ruan. “Genetic Algorithm–Driven Discovery of Unexpected Thermal Conductivity Enhancement by Disorder in Nanoporous Graphene”. In:Nano Energy71 (2020), p. 104619. DOI:10.1016/j.nanoen.2020.104619

    arXiv 2020

  54. [54]

    A review of uncertainty quantification in Deep learning: Techniques, applications and challenges

    Moloud Abdar et al. “A review of uncertainty quantification in Deep learning: Techniques, applications and challenges”. In:Information Fusion76 (2021), pp. 243–297.DOI: 10.1016/j.inffus.2021.05. 008

    work page doi:10.1016/j.inffus.2021.05 2021

  55. [55]

    Physics-Informed Neural Networks for Heat Transfer Problems

    S. Cai et al. “Physics-Informed Neural Networks for Heat Transfer Problems”. In:Journal of Heat Transfer143.6 (2021), p. 060801.DOI:10.1115/1.4050542

    doi:10.1115/1.4050542 2021

  56. [56]

    Non-Fourier phonon heat conduction at the microscale and nanoscale

    G. Chen. “Non-Fourier phonon heat conduction at the microscale and nanoscale”. In:Nature Reviews Physics3.8 (2021), pp. 555–569.URL: https://www.nature.com/articles/s42254-021-00334- 1

    2021

  57. [57]

    Deep Learning for the Modeling and Inverse Design of Radiative Heat Transfer

    Juan José García-Esteban, Jorge Bravo-Abad, and Juan Carlos Cuevas. “Deep Learning for the Modeling and Inverse Design of Radiative Heat Transfer”. In:arXiv preprint arXiv:2109.03114(2021)

    arXiv 2021

  58. [58]

    A survey of deep meta-learning

    Mike Huisman, Jan N Van Rijn, and Aske Plaat. “A survey of deep meta-learning”. In:Artificial Intelligence Review54.6 (2021), pp. 4483–4541

    2021

  59. [59]

    Highly accurate protein structure prediction with AlphaFold

    John Jumper et al. “Highly accurate protein structure prediction with AlphaFold”. In:Nature596 (2021), pp. 583–589.DOI:10.1038/s41586-021-03819-2

    doi:10.1038/s41586-021-03819-2 2021

  60. [60]

    Physics-informed machine learning

    George Em Karniadakis et al. “Physics-informed machine learning”. In:Nature Reviews Physics3.6 (2021), pp. 422–440

    2021

  61. [61]

    Physics-informed machine learning

    George Em Karniadakis et al. “Physics-informed machine learning”. In:Nature Reviews Physics3.6 (2021), pp. 422–440.DOI:10.1038/s42254-021-00314-5

    doi:10.1038/s42254-021-00314-5 2021

  62. [62]

    Ruiyang Li, Eungkyu Lee, and Tengfei Luo.Physics-Informed Neural Networks for Solving Multiscale Mode-Resolved Phonon Boltzmann Transport Equation. 2021. arXiv:2103.07983 [physics.comp-ph]. URL:https://arxiv.org/abs/2103.07983

    Pith/arXiv arXiv 2021

  63. [63]

    Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

    Lu Lu et al. “Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators”. In:Nature Machine Intelligence3.3 (2021), pp. 218–229.DOI: 10.1038/s42256-021- 00302-5. 21

    doi:10.1038/s42256-021- 2021

  64. [64]

    Physics-informed neural networks for nonlinear partial differential equations

    Lu Lu et al. “Physics-informed neural networks for nonlinear partial differential equations”. In:Commu- nications in Computational Physics28.1 (2021), pp. 1–48.DOI:10.4208/cicp.OA-2020-0193

    doi:10.4208/cicp.oa-2020-0193 2021

  65. [65]

    Component-wise physics-informed neural networks for PDEs

    Molly McBane et al. “Component-wise physics-informed neural networks for PDEs”. In:arXiv preprint arXiv:2107.12674(2021)

    Pith/arXiv arXiv 2021

  66. [66]

    Efficient calculations of the mode-resolved ab-initio thermal conductivity in nanos- tructures

    Giuseppe Romano. “Efficient calculations of the mode-resolved ab-initio thermal conductivity in nanos- tructures”. In:arXiv preprint arXiv:2105.08181(2021)

    Pith/arXiv arXiv 2021

  67. [67]

    OpenBTE: a solver for ab-initio phonon transport in multidimensional structures

    Giuseppe Romano. “OpenBTE: a solver for ab-initio phonon transport in multidimensional structures”. In:arXiv preprint arXiv:2106.02764(2021)

    Pith/arXiv arXiv 2021

  68. [68]

    Learning to correct physics simulators

    Gideon Dresdner et al. “Learning to correct physics simulators”. In:Nature Machine Intelligence4.8 (2022), pp. 730–739.DOI:10.1038/s42256-022-00506-9

    work page doi:10.1038/s42256-022-00506-9 2022

  69. [69]

    Physics-informed graph neural networks for modeling dynamics on irregular domains

    Han Gao and Lidong Sun. “Physics-informed graph neural networks for modeling dynamics on irregular domains”. In:Journal of Computational Physics449 (2022), p. 110754.DOI: 10.1016/j.jcp.2021. 110754

    doi:10.1016/j.jcp.2021 2022

  70. [70]

    Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium

    Ruiyang Li et al. “Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium”. In:npj Computational Materials8.1 (2022), p. 29

    2022

  71. [71]

    Multifidelity Deep Neural Operators for Efficient Learning of Partial Differential Equations with Application to Fast Inverse Design of Nanoscale Heat Transport

    L. Lu et al. “Multifidelity Deep Neural Operators for Efficient Learning of Partial Differential Equations with Application to Fast Inverse Design of Nanoscale Heat Transport”. In:Physical Review Research4.2 (2022).DOI:10.1103/physrevresearch.4.023210

    doi:10.1103/physrevresearch.4.023210 2022

  72. [72]

    FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators

    Jaideep Pathak et al. “FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators”. In:arXiv preprint arXiv:2202.11214(2022)

    Pith/arXiv arXiv 2022

  73. [73]

    Inverse Design in Nanoscale Heat Transport via Interpolating Interfacial Phonon Transmission

    G. Romano and S. G. Johnson. “Inverse Design in Nanoscale Heat Transport via Interpolating Interfacial Phonon Transmission”. In:Structural and Multidisciplinary Optimization65.10 (2022).DOI: 10.1007/ s00158-022-03392-w

    2022

  74. [74]

    Physics-informed neural networks with hard constraints for inverse design

    Peter Baddoo et al. “Physics-informed neural networks with hard constraints for inverse design”. In: arXiv preprint arXiv:2301.12345(2023)

    Pith/arXiv arXiv 2023

  75. [75]

    Physics-Informed Deep Neural Network for Inverse Heat Transfer Problems in Materials

    Md Muhtasim Billah et al. “Physics-Informed Deep Neural Network for Inverse Heat Transfer Problems in Materials”. In:Materials Today Communications35 (2023), p. 106336.DOI: 10.1016/j.mtcomm. 2023.106336

    arXiv 2023

  76. [76]

    Fast and accurate machine learning prediction of phonon scattering rates and lattice thermal conductivity

    Ziqi Guo et al. “Fast and accurate machine learning prediction of phonon scattering rates and lattice thermal conductivity”. In:npj Computational Materials9 (2023), p. 95.DOI: 10.1038/s41524-023- 01020-9

    work page doi:10.1038/s41524-023- 2023

  77. [77]

    Reduced-order model to predict thermal conductivity of dimensionally confined materials

    S Aria Hosseini, Alex Greaney, and Giuseppe Romano. “Reduced-order model to predict thermal conductivity of dimensionally confined materials”. In:Applied Physics Letters122.26 (2023)

    2023

  78. [78]

    Comparison of the performances between the gray and non-gray media approaches of thermal transport in silicon-tin

    Chukwunweike Iheduru et al. “Comparison of the performances between the gray and non-gray media approaches of thermal transport in silicon-tin”. In:Annals of Mathematics and Physics6.1 (2023), pp. 089–092.DOI:10.17352/amp.000085

    work page doi:10.17352/amp.000085 2023

  79. [79]

    Neural operator: Learning maps between function spaces

    Nikola B. Kovachki et al. “Neural operator: Learning maps between function spaces”. In:Journal of Machine Learning Research24.89 (2023), pp. 1–97

    2023

  80. [80]

    Physics-informed deep learning for solving coupled elec- tron and phonon Boltzmann transport equations

    Ruiyang Li, Eungkyu Lee, and Tengfei Luo. “Physics-informed deep learning for solving coupled elec- tron and phonon Boltzmann transport equations”. In:Physical Review Applied19.6 (2023), p. 064049. 22

    2023

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.