arxiv: 2604.03411 · v1 · submitted 2026-04-03 · 💻 cs.CE · cond-mat.soft

Recognition: 1 theorem link

· Lean Theorem

A Differentiable Framework for Gradient Enhanced Damage with Physics-Augmented Neural Networks in JAX-FEM

Adrian Buganza Tepole, Amirhossein Amiri-Hezaveh, Mark Wilkinson

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:05 UTC · model grok-4.3

classification 💻 cs.CE cond-mat.soft

keywords damage modelingneural networksfinite element methodsoft materialsgradient-enhanced damagephysics-augmented modelsconstitutive modelingmesh independence

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The pith

Input-convex neural networks combined with gradient-enhanced damage enable mesh-independent simulations of soft material degradation.

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

The paper develops a framework that parameterizes both the elastic strain energy and the damage yield function using input-convex neural networks inside the differentiable finite element code JAX-FEM. These networks are constructed to enforce polyconvexity and satisfaction of the Clausius-Duhem inequality automatically. An additional partial differential equation introduces a non-local damage field that regularizes the spatial distribution of damage and removes dependence on the finite element mesh. Validation covers local stress-strain data fits, single-element tests, uniform deformation studies, and a notched-plate example under heterogeneous loading. The approach therefore supplies a data-driven yet physically consistent route to damage modeling for rubbers, hydrogels, and biological tissues.

Core claim

The elastic strain energy and damage yield function are each represented by input-convex neural networks that guarantee polyconvexity and thermodynamic consistency by construction; these networks are embedded in a gradient-enhanced damage model whose non-local field is governed by an auxiliary PDE, yielding mesh-independent finite-element solutions for strain-softening in soft materials.

What carries the argument

Input-convex neural networks (ICNNs) that enforce polyconvexity of the strain-energy function and satisfaction of the Clausius-Duhem inequality by design, together with the non-local damage variable introduced through an auxiliary gradient-enhanced partial differential equation.

If this is right

Damage simulations become possible with constitutive models learned directly from experimental data while remaining thermodynamically consistent.
Mesh sensitivity disappears in strain-softening problems once the non-local damage PDE is solved alongside the mechanical equilibrium equations.
The fully differentiable implementation supports automatic differentiation for parameter calibration and design optimization tasks.
Open-source release of the JAX-FEM code reduces the effort required to apply physics-augmented neural networks to new soft-material problems.

Where Pith is reading between the lines

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

The same architecture could be extended to couple damage with viscoelasticity or growth by adding further network outputs and auxiliary fields.
Because the framework is differentiable, material parameters inside the networks could be optimized directly against full-field experimental measurements.
Transfer of the trained networks across different soft materials may be feasible if the ICNN structure preserves the required convexity properties under modest retraining.

Load-bearing premise

The input-convex neural networks trained on limited data will continue to enforce polyconvexity and the Clausius-Duhem inequality for all unseen loading paths, and the added non-local PDE will fully remove mesh dependence in three-dimensional heterogeneous geometries.

What would settle it

Observation of continued mesh-dependent damage localization in a three-dimensional heterogeneous notched specimen, or violation of thermodynamic consistency in the stress response along a loading path outside the training set.

Figures

Figures reproduced from arXiv: 2604.03411 by Adrian Buganza Tepole, Amirhossein Amiri-Hezaveh, Mark Wilkinson.

**Figure 1.** Figure 1: Single-element response under uniaxial loading. a) Axial stress 𝜎11 vs. stretch. b) Local damage variable 𝜅 vs. stretch, c) Damage state 𝑑 = 1 − 𝑓𝑑 (𝜅) vs. stretch. with respect to these parameters. This is because when the non-local regularization is disabled, the local damage variable 𝜅 is equal to the non-local damage variable 𝜙 causing Eq. (37) to reduce exclusively to the neo-Hookean strain-energy den… view at source ↗

**Figure 2.** Figure 2: Data-driven model of continuum damage. The same architecture was fitted to three sets of data (a-c), resulting in a pair of physics-augmented neural networks for each material. were gathered from published damage models in the literature (Amiri-Hezaveh and Tepole, 2025). As described in the Methods, the data-driven constitutive model is specified by two ICNNs, one for the isochoric strain energy 𝜓𝑖𝑠𝑜 and … view at source ↗

**Figure 3.** Figure 3: Comparison of the staggered and monolithic solution schemes for the uniform deformation and mesh independence study. Simulations terminated when damage variable reached 𝑑 = 0.995 or the solver failed to converge. are those from the data-driven model in Fig. 2b. The constants employed for this example are the same as those used in the previous section, except that 𝜂𝑑 = 0.001 MPa−1 is selected. For the Newt… view at source ↗

**Figure 4.** Figure 4: Damage field 𝑑 at a prescribed displacement of 𝑢𝑥 = 0.75mm. Simulations terminated at 𝑑 = 0.995. (a) fails to capture damage evolution effectively, (b) shows expected damage but localizes due to mesh selection, (c) avoids localization while capturing expected damage evolution. a b c ! " #!" [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Notched plate simulations at three increment frames of the simulation with a final 25 mm displacement on the right boundary. a) Global damage field 𝜙, b) local damage degradation 𝑑, c) Von Mises stress. incrementation step sensitivity analysis, the three cases are investigated: 1) 25 steps, 2) 50 steps, and 3) 100 steps [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Notched plate simulations. a) Convergence with respect to the number of increment steps. b) Convergence with respect to mesh refinement. 6. Discussion and conclusion In this work we presented a finite element implementation of a data-driven gradient-enhanced damage model within the open-source package JAX-FEM. The implementation brings together two recent developments in scientific machine learning: phys… view at source ↗

read the original abstract

Soft materials such as rubbers, hydrogels, and biological tissues undergo damage in the form of stiffness degradation without apparent changes in their stress-free geometry. Accurate simulation of this behavior is critical in applications ranging from soft robotics to the design of medical devices, yet two persistent challenges are the difficulty of constructing flexible, thermodynamically consistent constitutive models, and the mesh dependence of finite element solutions caused by strain softening. Here we address both challenges simultaneously by combining physics-augmented neural network constitutive models with a gradient-enhanced damage formulation implemented within the differentiable finite element framework JAX-FEM. The elastic strain energy and the damage yield function are each parameterized by input-convex neural networks (ICNNs), which enforce polyconvexity and satisfaction of the Clausius--Duhem inequality by design. The gradient-enhanced formulation introduces a non-local damage field governed by an additional partial differential equation, regularizing the spatial distribution of damage and eliminating mesh dependence. The implementation is validated through local stress-strain fits, single-element parametric studies, a mesh and solution strategy study for a uniform deformation case, and a notched plate simulation. The results demonstrate that the proposed framework enables flexible, data-driven, mesh-independent damage simulation for a broad class of soft materials. We anticipate that the open-source implementation will lower the barrier to adopting physics-augmented neural network constitutive models.

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

The paper gives a working open-source JAX-FEM template that wires ICNNs into both the strain-energy and yield functions inside a gradient-enhanced damage model, which is new enough to check out if you need data-driven softening in soft solids.

read the letter

The core contribution is the concrete pairing of input-convex networks for the elastic energy and the damage yield surface with a non-local damage PDE, all inside the differentiable JAX-FEM solver. That specific stack has not been published before in the references they list. The architecture choice is sensible: ICNNs enforce polyconvexity and the dissipation inequality by construction, so no extra loss terms are needed, and the gradient regularization on the damage field is the standard fix for mesh sensitivity in softening problems. They supply the code, which is the part that actually matters for reuse in soft robotics or tissue modeling workflows. The test suite covers local stress-strain calibration, single-element parametric checks, a uniform mesh-convergence study, and a notched-plate example, which is a fair set of demonstrations for this style of paper. The implementation appears reproducible because everything runs end-to-end in JAX. The main limitation is that the reported results stay qualitative. No error tables, no convergence rates against analytical solutions, and no explicit check that the learned functions remain consistent once the non-local damage field drives the material points outside the original training distribution. In heterogeneous 3-D geometries that is the regime that actually matters, so the extrapolation claim rests on the architecture alone rather than on shown evidence. Readers who already use neural constitutive models or JAX-based FEM will get immediate value from the code and the template. The work is coherent on its own terms and deserves a serious referee who can ask for quantitative metrics and at least one out-of-distribution loading test. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a differentiable framework that parameterizes the elastic strain-energy density and damage yield function using input-convex neural networks (ICNNs) within a gradient-enhanced damage formulation, all implemented in the JAX-FEM code. The ICNN architecture is intended to enforce polyconvexity and thermodynamic consistency (Clausius-Duhem) by construction, while the non-local damage PDE regularizes the problem to remove mesh dependence. Validation consists of local stress-strain fits, single-element parametric studies, a mesh/solution-strategy study on uniform deformation, and a notched-plate example; the central claim is that the approach enables flexible, data-driven, mesh-independent damage simulation for soft materials.

Significance. If the validation evidence were quantitative and the out-of-distribution consistency were demonstrated, the work would offer a useful bridge between data-driven constitutive modeling and physics-constrained finite-element simulation. The open-source JAX-FEM implementation and the simultaneous treatment of constitutive flexibility and numerical regularization would be genuine strengths for the computational mechanics community working on soft materials.

major comments (2)

[Abstract / validation studies] Abstract and validation description: the manuscript states that validation was performed via stress-strain fits, parametric studies, mesh studies, and a notched-plate example, yet reports no quantitative error metrics, convergence rates, or comparisons against analytical solutions. Without these, the claims of accuracy and mesh independence remain unsubstantiated and constitute a load-bearing gap for the central contribution.
[ICNN constitutive model section] ICNN parameterization and consistency enforcement: the architecture guarantees polyconvexity and Clausius-Duhem satisfaction only inside the convex hull of the (limited) training data. No systematic verification is provided that the learned functions continue to satisfy the required inequalities on out-of-distribution paths that arise once the non-local damage PDE couples fields across a heterogeneous 3-D mesh.

minor comments (2)

[Abstract] The abstract would be strengthened by indicating the volume and type of training data used for the ICNNs and the specific soft-material classes considered.
[Governing equations] Notation for the non-local damage variable and its coupling to the local damage field should be made fully explicit in the governing equations to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the validation requirements and consistency aspects of our framework. We address each major comment below.

read point-by-point responses

Referee: [Abstract / validation studies] Abstract and validation description: the manuscript states that validation was performed via stress-strain fits, parametric studies, mesh studies, and a notched-plate example, yet reports no quantitative error metrics, convergence rates, or comparisons against analytical solutions. Without these, the claims of accuracy and mesh independence remain unsubstantiated and constitute a load-bearing gap for the central contribution.

Authors: We agree that quantitative metrics would strengthen the substantiation of accuracy and mesh independence. In the revised manuscript we will add explicit error metrics, including relative L2 errors between predicted and reference stress-strain curves for the local fits, convergence rates (error versus element size) for the uniform deformation mesh study, and direct comparisons to analytical solutions for the single-element parametric studies. For the notched-plate example we will report quantitative indicators of mesh independence such as variation in peak reaction force and damage-zone width across successively refined meshes. revision: yes
Referee: [ICNN constitutive model section] ICNN parameterization and consistency enforcement: the architecture guarantees polyconvexity and Clausius-Duhem satisfaction only inside the convex hull of the (limited) training data. No systematic verification is provided that the learned functions continue to satisfy the required inequalities on out-of-distribution paths that arise once the non-local damage PDE couples fields across a heterogeneous 3-D mesh.

Authors: The ICNN architecture enforces polyconvexity and Clausius-Duhem satisfaction globally by construction (non-negative weights and convex activations) for all inputs, independent of the training-data convex hull; the training data only governs approximation fidelity inside the sampled region. We acknowledge that no explicit out-of-distribution verification was included for paths arising in the coupled 3-D simulations. In the revision we will add a dedicated verification subsection that extracts representative strain histories from the notched-plate simulation (including paths outside the original training hull) and confirms that the required inequalities continue to hold. revision: partial

Circularity Check

0 steps flagged

No significant circularity; modeling choices and architectural constraints are independent of fitted outputs

full rationale

The paper describes an implementation framework that parameterizes strain energy and yield functions via input-convex neural networks chosen specifically to enforce polyconvexity and the Clausius-Duhem inequality by architecture, then couples this to a standard gradient-enhanced non-local damage PDE inside JAX-FEM. These are explicit design decisions rather than derivations in which a claimed prediction reduces to a fitted parameter or self-citation by construction. No equation equates an output quantity directly to an input fit; validation proceeds through separate local fits, single-element tests, and mesh studies whose success is not presupposed by the constitutive choice itself. The central result (mesh-independent data-driven damage simulation) therefore rests on independent numerical evidence and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to statements explicitly present in the provided text.

free parameters (1)

ICNN weights for strain-energy and yield functions
Neural-network parameters are fitted to data; their specific values are not reported in the abstract.

axioms (2)

domain assumption Input-convex neural networks enforce polyconvexity and satisfaction of the Clausius-Duhem inequality by design
Stated directly in the abstract as the mechanism guaranteeing thermodynamic consistency.
domain assumption The gradient-enhanced damage PDE regularizes the spatial distribution of damage and eliminates mesh dependence
Presented as the solution to the mesh-dependence problem without further proof in the abstract.

pith-pipeline@v0.9.0 · 5551 in / 1352 out tokens · 47911 ms · 2026-05-13T18:05:22.958774+00:00 · methodology

discussion (0)

Reference graph

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