arxiv: 2603.29184 · v3 · submitted 2026-03-31 · 💻 cs.LG · cs.NA· math.NA

Recognition: unknown

Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks

Anci Lin , Zhiwen Zhang , Wenju Zhao

Authors on Pith no claims yet

Pith reviewed 2026-05-14 00:28 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA

keywords biomimetic physics-informed neural networkscell-induced phase transitionsextracellular matricesdensificationtether formationnonconvex multi-well energiestransition layers

0 comments

The pith

Biomimetic physics-informed neural networks recover cell-induced densification and tethers more reliably than standard adaptive baselines.

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

The paper tackles numerical difficulties posed by fine-scale microstructures, low-regularity layers, and sharp interfaces that emerge from nonconvex multi-well energies during cell-induced phase transitions in fibrous extracellular matrices. It introduces biomimetic physics-informed neural networks that apply a near-to-far curriculum, progressively exposing the domain starting from the cell boundary, together with a deformation-uncertainty proxy to concentrate collocation points on transition layers and tether-forming zones. Benchmarks on single-cell and multicellular cases show these networks recover the densified phase near boundaries and gaps while reproducing tether shapes more accurately than ungated or residual-driven adaptive methods. Such modeling improvements matter because they enable better simulation of how cells remodel their surrounding matrix, with direct relevance to tissue mechanics and cell behavior.

Core claim

Bio-PINNs implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combine this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.

What carries the argument

The near-to-far curriculum schedule combined with a deformation-uncertainty proxy inside Bio-PINNs, which expands the solved domain outward from the cell and adaptively places points to resolve low-regularity features in the nonconvex energy landscape.

If this is right

Densified phases are recovered more reliably near cell boundaries and intercellular gaps than with ungated or residual-driven methods.
Tether morphologies are captured with higher fidelity in both single-cell and multicellular geometries.
Low-regularity transition layers in nonconvex energies become tractable without manual mesh refinement.
The same curriculum-plus-proxy strategy applies directly to related phase-transition problems that produce sharp interfaces.

Where Pith is reading between the lines

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

The approach may extend to other sharp-interface problems in materials science that share nonconvex energy structures.
Higher accuracy in tether and densification predictions could improve downstream models of cell migration or tissue remodeling.
Direct comparison against time-lapse imaging of matrix tethers would provide an external test of whether the learned solutions match physical observations.

Load-bearing premise

The near-to-far curriculum combined with the deformation-uncertainty proxy will reliably concentrate collocation points on evolving transition layers without introducing bias or missing other low-regularity features in the nonconvex multi-well energy landscape.

What would settle it

A controlled single-cell benchmark in which high-resolution reference solutions show a known tether or densification pattern that Bio-PINNs systematically miss or distort while baselines do not.

read the original abstract

Nonconvex multi-well energies in cell-induced phase transitions give rise to fine-scale microstructures, low-regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics-informed learning. Here we introduce biomimetic physics-informed neural networks (Bio-PINNs), which implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Bio-PINNs pair a near-to-far curriculum with a deformation-uncertainty proxy to target transition layers in cell-matrix simulations, but the abstract supplies no numbers to show the gains are real.

read the letter

The paper's main move is to add a progressive near-to-far domain reveal starting at the cell boundary together with a deformation-uncertainty proxy that steers collocation points toward evolving tethers and densified zones. This combination is presented as new for this class of nonconvex multi-well problems in fibrous ECM modeling. It directly tackles the numerical difficulty of sharp interfaces and fine microstructures that standard PINNs struggle with in single-cell and multicellular setups. The claim is that the method recovers the densified phase more reliably near boundaries and intercellular gaps and preserves tether morphology better than ungated or residual-driven adaptive baselines. That targeted improvement fits the biophysical setting and could matter for tissue engineering simulations where those features drive mechanical behavior. The abstract does not include any error metrics, ablation results, or details on the proxy weighting and curriculum schedule, so the size of the improvement stays unclear. The stress-test point about possible under-sampling of distant or secondary interfaces in the nonconvex landscape is reasonable to raise; without the full implementation it is hard to judge whether the proxy definition avoids that bias. The work is aimed at researchers already using PINNs for phase-transition problems in biophysics or mechanics. A reader who needs a practical way to concentrate points on low-regularity layers might find the curriculum idea worth testing. The paper deserves peer review so that referees can examine the quantitative benchmarks and the exact form of the uncertainty proxy.

Referee Report

2 major / 1 minor

Summary. The paper introduces biomimetic physics-informed neural networks (Bio-PINNs) for simulating cell-induced phase transitions in fibrous extracellular matrices governed by nonconvex multi-well energies. It proposes a near-to-far curriculum that progressively reveals the domain starting from cell boundaries, combined with a deformation-uncertainty proxy to adaptively concentrate collocation points on transition layers and tether-forming regions. The central claim is that Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps while capturing tether morphology more faithfully than ungated and residual-driven adaptive baselines across single-cell and multicellular benchmarks.

Significance. If the quantitative evidence supports the claims, the work could advance physics-informed learning for problems with fine-scale microstructures and sharp interfaces in biological mechanics, providing a practical way to handle low-regularity features without excessive computational cost. The biomimetic curriculum and proxy represent a targeted adaptation that may generalize to other nonconvex energy landscapes.

major comments (2)

[Abstract] Abstract: The abstract asserts superior recovery of densified phases and tether morphology on benchmarks, but supplies no quantitative metrics, error bars, ablation studies, or derivation details, leaving the central claim without visible supporting evidence.
[Curriculum and proxy] Curriculum description: The assumption that the near-to-far curriculum combined with the deformation-uncertainty proxy reliably concentrates points on all evolving transition layers without bias is load-bearing for the intercellular-gap and tether claims; in a nonconvex multi-well landscape, secondary interfaces can nucleate away from initial cell-boundary regions, and no explicit test (e.g., sampling-density maps or missed-feature diagnostics) is referenced to rule this out.

minor comments (1)

[Methods] Clarify the precise mathematical definition of the deformation-uncertainty proxy (including its weighting relative to residual norms) and list the free parameters of the revealing schedule so that the method is reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. We address each of the major comments below and have made revisions to strengthen the presentation of our results and methods.

read point-by-point responses

Referee: [Abstract] Abstract: The abstract asserts superior recovery of densified phases and tether morphology on benchmarks, but supplies no quantitative metrics, error bars, ablation studies, or derivation details, leaving the central claim without visible supporting evidence.

Authors: We agree that the abstract would benefit from including quantitative support for the claims. The full manuscript provides detailed quantitative comparisons, including error metrics with standard deviations and ablation studies in the results sections. To address this, we have revised the abstract to incorporate key quantitative results from the benchmarks. revision: yes
Referee: [Curriculum and proxy] Curriculum description: The assumption that the near-to-far curriculum combined with the deformation-uncertainty proxy reliably concentrates points on all evolving transition layers without bias is load-bearing for the intercellular-gap and tether claims; in a nonconvex multi-well landscape, secondary interfaces can nucleate away from initial cell-boundary regions, and no explicit test (e.g., sampling-density maps or missed-feature diagnostics) is referenced to rule this out.

Authors: We appreciate the referee's concern about potential bias in the curriculum and proxy for capturing all transition layers in nonconvex landscapes. The deformation-uncertainty proxy is formulated to identify regions with high predictive uncertainty in the deformation field, which includes both primary and secondary interfaces. Our benchmarks demonstrate successful recovery of intercellular gaps and tethers. To provide explicit verification, we have included sampling-density maps and missed-feature diagnostics in the revised supplementary material. revision: yes

Circularity Check

0 steps flagged

Bio-PINNs near-to-far curriculum and deformation-uncertainty proxy are independent innovations evaluated against external baselines

full rationale

The paper introduces Bio-PINNs as a new architecture that combines a progressive near-to-far domain reveal with a deformation-uncertainty proxy for collocation-point concentration. These elements are presented as novel contributions whose performance is measured on single-cell and multicellular benchmarks against ungated and residual-driven adaptive baselines. No equation or claim reduces by construction to a fitted parameter or self-citation; the central claims rest on comparative empirical recovery of densified phases and tether morphology rather than tautological re-derivation of the inputs. This is the normal case of a self-contained methodological paper whose innovations are externally falsifiable.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the effectiveness of two newly introduced mechanisms (curriculum schedule and uncertainty proxy) whose parameters and convergence properties are not detailed in the abstract; standard PINN approximation assumptions are also invoked without explicit verification.

free parameters (2)

curriculum revealing schedule parameters
Parameters controlling the progressive expansion of the computational domain away from the cell boundary; these must be chosen or tuned to achieve the reported behavior.
deformation-uncertainty proxy weighting
Scaling factor that determines how strongly collocation points are concentrated near transition layers and tether regions.

axioms (2)

standard math Neural networks can approximate weak solutions to nonconvex multi-well variational problems with sufficient accuracy when collocation points are appropriately placed
Core assumption underlying all PINN methods for this class of PDEs.
ad hoc to paper The near-to-far curriculum does not introduce systematic bias into the learned solution of the cell-induced phase transition
Specific to the Bio-PINN design and not justified in the abstract.

invented entities (1)

Bio-PINNs no independent evidence
purpose: A physics-informed neural network variant that combines curriculum learning with uncertainty-driven adaptive sampling for cell-matrix problems
The new architecture introduced in the paper.

pith-pipeline@v0.9.0 · 5421 in / 1529 out tokens · 44861 ms · 2026-05-14T00:28:34.052677+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Journal of the Royal Society Interface18(175), 20200823 (2021) https://doi.org/10.1098/rsif.2020.0823

Grekas, G., Proestaki, M., Rosakis, P., Notbohm, J., Makridakis, C., Ravichan- dran, G.: Cells exploit a phase transition to mechanically remodel the fibrous extracellular matrix. Journal of the Royal Society Interface18(175), 20200823 (2021) https://doi.org/10.1098/rsif.2020.0823

work page doi:10.1098/rsif.2020.0823 2021
[2]

SIAM Journal on Numerical Analysis60(2), 715–750 (2022) https://doi.org/10.1137/20M137286X

Grekas, G., Koumatos, K., Makridakis, C., Rosakis, P.: Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials:γ- convergence analysis. SIAM Journal on Numerical Analysis60(2), 715–750 (2022) https://doi.org/10.1137/20M137286X

work page doi:10.1137/20m137286x 2022
[3]

Archive for Rational Mechanics and Analysis63(4), 337–403 (1976) https://doi

Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis63(4), 337–403 (1976) https://doi. org/10.1007/BF00279992

work page doi:10.1007/bf00279992 1976
[4]

Archive for Rational Mechanics and Analysis115, 329–365 (1991) https://doi.org/10.1007/BF00375129

Kinderlehrer, D., Pedregal, P.: Characterizations of young measures generated by gradients. Archive for Rational Mechanics and Analysis115, 329–365 (1991) https://doi.org/10.1007/BF00375129

work page doi:10.1007/bf00375129 1991
[5]

Communications on Pure and Applied Mathematics39(1), 113–137 (1986) https://doi.org/10.1002/cpa.3160390107

Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems, I. Communications on Pure and Applied Mathematics39(1), 113–137 (1986) https://doi.org/10.1002/cpa.3160390107

work page doi:10.1002/cpa.3160390107 1986
[6]

Proceedings of the National Academy of Sciences 111(2), 658–663 (2014) https://doi.org/10.1073/pnas.1311312110 27

Shi, Q., Ghosh, R.P., Engelke, H., Rycroft, C.H., Cassereau, L., Sethian, J.A., Weaver, V.M., Liphardt, J.T.: Rapid disorganization of mechanically interacting systems of mammary acini. Proceedings of the National Academy of Sciences 111(2), 658–663 (2014) https://doi.org/10.1073/pnas.1311312110 27

work page doi:10.1073/pnas.1311312110 2014
[7]

SIAM Journal on Numerical Analysis39(5), 1749–1779 (2002) https://doi.org/10.1137/S0036142901384162

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of dis- continuous galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis39(5), 1749–1779 (2002) https://doi.org/10.1137/S0036142901384162

work page doi:10.1137/s0036142901384162 2002
[8]

Journal of Scientific Computing 22(1–3), 83–118 (2005) https://doi.org/10.1007/s10915-004-4135-7

Brenner, S.C., Sung, L.-Y.:c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. Journal of Scientific Computing 22(1–3), 83–118 (2005) https://doi.org/10.1007/s10915-004-4135-7

work page doi:10.1007/s10915-004-4135-7 2005
[9]

The Aeronautical Journal72(692), 701–709 (1968) https://doi.org/10.1017/S000192400008489X

Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. The Aeronautical Journal72(692), 701–709 (1968) https://doi.org/10.1017/S000192400008489X

work page doi:10.1017/s000192400008489x 1968
[10]

Studies in Mathematics and its Applications, vol

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1978)

work page 1978
[11]

Computer Methods in Applied Mechanics and Engineering194(39–41), 4135–4195 (2005) https://doi

Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering194(39–41), 4135–4195 (2005) https://doi. org/10.1016/j.cma.2004.10.008

work page doi:10.1016/j.cma.2004.10.008 2005
[12]

Springer Series in Computational Mathematics, vol

Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York, NY (1991). https:// doi.org/10.1007/978-1-4612-3172-1 .https://doi.org/10.1007/978-1-4612-3172-1

work page doi:10.1007/978-1-4612-3172-1 1991
[13]

SIAM Studies in Applied Mathematics, vol

Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics, Philadelphia, PA (1989). https: //doi.org/10.1137/1.9781611970838 .https://doi.org/10.1137/1.9781611970838

work page doi:10.1137/1.9781611970838 1989
[14]

Springer Series in Computational Mathematics, vol

Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Intro- duction. Springer Series in Computational Mathematics, vol. 13. Springer, Berlin, Heidelberg (1990). https://doi.org/10.1007/978-3-642-61257-2 . https://doi.org/10.1007/978-3-642-61257-2

work page doi:10.1007/978-3-642-61257-2 1990
[15]

Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. interfacial free energy. The Journal of Chemical Physics28(2), 258–267 (1958) https://doi.org/ 10.1063/1.1744102

work page doi:10.1063/1.1744102 1958
[16]

A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening

Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica27(6), 1085–1095 (1979) https://doi.org/10.1016/0001-6160(79)90196-2

work page doi:10.1016/0001-6160(79)90196-2 1979
[18]

Springer, New York, NY (2002)

Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York, NY (2002). https://doi.org/10.1007/b97697 .https://doi.org/10.1007/b97697 28

work page doi:10.1007/b97697 2002
[19]

Raissi, A

Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science367(6481), 1026– 1030 (2020) https://doi.org/10.1126/science.aaw4741

work page doi:10.1126/science.aaw4741 2020
[20]

Journal of Computational Physics426, 109951 (2021) https://doi.org/10.1016/ j.jcp.2020.109951

Jin, X., Cai, S., Li, H., Karniadakis, G.E.: Nsfnets (navier–stokes flow nets): Physics-informed neural networks for the incompressible navier–stokes equations. Journal of Computational Physics426, 109951 (2021) https://doi.org/10.1016/ j.jcp.2020.109951

work page arXiv 2021
[21]

Water Resources Research57(7), 2020–029479 (2021) https://doi.org/10.1029/2020WR029479

He, Q., Tartakovsky, A.M.: Physics-informed neural network method for forward and backward advection-dispersion equations. Water Resources Research57(7), 2020–029479 (2021) https://doi.org/10.1029/2020WR029479

work page doi:10.1029/2020wr029479 2020
[22]

Journal of Computational Physics425, 109913 (2021) https://doi.org/10.1016/j.jcp.2020

Yang, L., Meng, X., Karniadakis, G.E.: B-pinns: Bayesian physics-informed neu- ral networks for forward and inverse pde problems with noisy data. Journal of Computational Physics425, 109913 (2021) https://doi.org/10.1016/j.jcp.2020. 109913

work page doi:10.1016/j.jcp.2020 2021
[23]

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

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics378, 686–707 (2019) https://doi.org/10.1016/j.jcp.2018.10.045

work page doi:10.1016/j.jcp.2018.10.045 2019
[24]

Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang

Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nature Reviews Physics3(6), 422–440 (2021) https://doi.org/10.1038/s42254-021-00314-5

work page doi:10.1038/s42254-021-00314-5 2021
[25]

In: Advances in Neural Information Processing Systems, vol

Krishnapriyan, A.S., Gholami, A., Zhe, S., Kirby, R.M., Mahoney, M.W.: Charac- terizing possible failure modes in physics-informed neural networks. In: Advances in Neural Information Processing Systems, vol. 34, pp. 26548–26560 (2021). https://doi.org/10.48550/arXiv.2109.01050 .https://arxiv.org/abs/2109.01050

work page doi:10.48550/arxiv.2109.01050 2021
[26]

When and why PINNs fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

Wang, S., Yu, X., Perdikaris, P.: When and why pinns fail to train: A neural tangent kernel perspective. Journal of Computational Physics449, 110768 (2022) https://doi.org/10.1016/j.jcp.2021.110768

work page doi:10.1016/j.jcp.2021.110768 2022
[27]

In: Proceedings of the 40th International Conference on Machine Learn- ing

Daw, A., Bu, J., Wang, S., Perdikaris, P., Karpatne, A.: Mitigating propagation failures in physics-informed neural networks using retain-resample-release (R3) sampling. In: Proceedings of the 40th International Conference on Machine Learn- ing. Proceedings of Machine Learning Research, vol. 202, pp. 7264–7302 (2023). https://proceedings.mlr.press/v202/da...

work page 2023
[28]

In: Proceedings of the 36th International Conference on Machine Learning

Rahaman, N., Arpit, D., Baratin, A., Draxler, F., Lin, M., Hamprecht, F., Bengio, Y., Courville, A.: On the spectral bias of neural networks. In: Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 97, pp. 5301–5310 (2019). https://proceedings.mlr.press/v97/rahaman19a.html 29

work page 2019
[29]

Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Ragha- van, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T

Tancik, M., Srinivasan, P.P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J.T., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. In: Advances in Neural Information Processing Systems, vol. 33, pp. 7537–7547 (2020). https://doi.org/ 10.48550/arXiv.2006.10739 .https:...

work page doi:10.48550/arxiv.2006.10739 2020
[30]

The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

E, W., Yu, B.: The deep ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6(1), 1–12 (2018) https://doi.org/10.1007/s40304-018-0127-z

work page doi:10.1007/s40304-018-0127-z 2018
[31]

Computer Methods in Applied Mechanics and Engineering374, 113547 (2021) https://doi.org/10.1016/ j.cma.2020.113547 arXiv:2003.05385 [math.NA]

Kharazmi, E., Zhang, Z., Karniadakis, G.E.: hp-vpinns: Variational physics- informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering374, 113547 (2021) https://doi.org/10.1016/ j.cma.2020.113547 arXiv:2003.05385 [math.NA]

work page arXiv 2021
[32]

Archive for Rational Mechanics and Analysis100(1), 13–52 (1987) https://doi.org/10.1007/ BF00281246

Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis100(1), 13–52 (1987) https://doi.org/10.1007/ BF00281246

work page 1987
[33]

Proceedings of the Royal Society of Edinburgh: Section A Mathematics 88(3-4), 315–328 (1981) https://doi.org/10.1017/S030821050002014X

Ball, J.M.: Global invertibility of sobolev functions and the interpenetration of matter. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 88(3-4), 315–328 (1981) https://doi.org/10.1017/S030821050002014X

work page doi:10.1017/s030821050002014x 1981
[34]

Archive for Ratio- nal Mechanics and Analysis11, 385–414 (1962) https://doi.org/10.1007/ BF00253945

Toupin, R.A.: Elastic materials with couple-stresses. Archive for Ratio- nal Mechanics and Analysis11, 385–414 (1962) https://doi.org/10.1007/ BF00253945

work page 1962
[35]

Archive for Rational Mechanics and Analysis16, 51–78 (1964) https://doi.org/10.1007/BF00248490

Mindlin, R.D.: Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis16, 51–78 (1964) https://doi.org/10.1007/BF00248490

work page doi:10.1007/bf00248490 1964
[36]

IEEE Transactions on Neural Networks 9(5), 987–1000 (1998) https://doi.org/10.1109/72.712178

Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordi- nary and partial differential equations. IEEE Transactions on Neural Networks 9(5), 987–1000 (1998) https://doi.org/10.1109/72.712178

work page doi:10.1109/72.712178 1998
[37]

Estimating the D imension of a M odel

Efron, B.: Bootstrap methods: Another look at the jackknife. The Annals of Statistics7(1), 1–26 (1979) https://doi.org/10.1214/aos/1176344552

work page doi:10.1214/aos/1176344552 1979
[38]

In: Proceedings of the 33rd International Conference on Machine Learning

Gal, Y., Ghahramani, Z.: Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In: Proceedings of the 33rd International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 48, pp. 1050–1059 (2016).https://proceedings.mlr.press/v48/gal16.html

work page 2016
[39]

Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles

Lakshminarayanan, B., Pritzel, A., Blundell, C.: Simple and scalable predictive uncertainty estimation using deep ensembles. In: Advances in Neural Informa- tion Processing Systems, vol. 30, pp. 6402–6413 (2017). https://doi.org/10.48550/ arXiv.1612.01474 .https://arxiv.org/abs/1612.01474 30

work page Pith review arXiv 2017
[40]

Wu, C., Zhu, M., Tan, Q., Kartha, Y., Lu, L.: A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering403, 115671 (2023) https://doi.org/10.1016/j.cma.2022.115671 arXiv:2207.10289 [cs.LG] Acknowledgements Wenju Zhao was supported by the National K...

work page doi:10.1016/j.cma.2022.115671 2023