Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations

Elasto-plastic boundary value problems in geotechnical engineering are conventionally solved by the Finite Element Method (FEM), which incurs high computational cost from incremental-iterative procedures. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but suffer from spectral bias, failing to resolve the sharp gradients arising at elastic-plastic boundaries and within localized plastic zones. This limitation is particularly consequential for the non-associative Mohr-Coulomb model, whose pressure-dependent yield surface and dilatant flow rule generate narrower plastic zones and steeper stress gradients than pressure-independent criteria. This study proposes a Fourier Feature Physics-Informed Neural Network (FF-PINN) for two-dimensional elasto-plastic problems governed by this model. Random Fourier feature mapping is embedded into the input layer to mitigate spectral bias, supported by a multi-objective loss function enforcing equilibrium, constitutive relations, and Karush-Kuhn-Tucker conditions against high-fidelity FEM data, together with a strain-adaptive sampling strategy. Benchmarked across three test cases, FF-PINN achieves superior accuracy across most predicted fields, with error reductions up to approximately 66 percent in displacement and 27 percent in stress components, and reproduces the plastic failure zone geometry with markedly closer fidelity to FEM. Sensitivity analysis confirms robustness across training data size, collocation density, loss weighting, and noise levels up to 2.0 percent. FF-PINN converges in half the training epochs required by the conventional PINN, halving wall-clock training time while achieving greater predictive accuracy. The framework therefore offers a computationally efficient and physics-consistent alternative to FEM for elasto-plastic geotechnical analysis.

Constitutive artificial neural networks (CANNs) provide interpretable material model discovery, but have so far been used in stress-supervised settings based on apparent stress-strain data from homogeneous tests. Because each test samples only a narrow loading path and provides homogenized rather than local stress information, robust discovery typically requires multiple loading modes to constrain the multidimensional response. This is challenging for soft biological tissues, where repeated testing, damage, and sample variability limit reliable information from a single specimen. Here, we combine CANNs with the stress-unsupervised full-field discovery framework EUCLID to identify sparse hyperelastic laws directly from displacement fields and reaction forces in one heterogeneity-inducing loading case. CANN-EUCLID minimizes equilibrium imbalance with sparsity-promoting regularization selecting compact active terms, without local stress measurements or a prescribed law. We evaluate the approach on isotropic and anisotropic benchmarks with prescribed ground-truth laws. When the ground truth is representable by the chosen CANN basis, our method recovers the correct terms with near-exact accuracy, including exponential terms with embedded parameters. When it is not contained in the basis, the method retains shared terms and approximates missing contributions using available basis functions. Generalization depends strongly on sampled deformation states: exponential strain-stiffening terms can be recovered accurately when sufficiently probed, but can produce large extrapolation errors when the stiffening regime lies outside the sampled domain. Forward FE validation simulations show that the discovered behavior accurately replicates the ground truth. These results establish stress-unsupervised CANN discovery as a promising framework for interpretable full-field constitutive model identification.

The identification of constitutive neural network models from heterogeneous full-field deformation data provides a robust alternative to traditional calibration methods based on homogeneous stress-strain experiments, particularly given the high dimensionality of trainable parameters. Existing approaches must balance generality, robustness, and computational efficiency: Conventional finite element model updating is broadly applicable but computationally demanding; weak-form methods offer efficiency but are sensitive to noise and data scarcity; neural operator models are highly expressive but require extensive training datasets. This work presents FE-MAD (Finite Element-Based Material learning via Automatic Differentiation), an end-to-end differentiable framework that integrates a constitutive neural network model within a JAX-FEM nonlinear solver and identifies its parameters through gradient-based minimization of a measurement-mismatch loss. Newton tangent stiffness and loss gradients are computed automatically using forward- and reverse-mode automatic differentiation throughout the entire pipeline, thereby removing the need for analytic adjoints or offline surrogate models. FE-MAD is demonstrated for two architectures: a grey-box Constitutive Artificial Neural Network (CANN), a polyconvex, fully connected model with high flexibility, and a white-box CANN, an expert-system network with phenomenologically interpretable strain-energy terms. Focusing on incompressible isotropic hyperelasticity, FE-MAD is evaluated on three open experimental datasets: (1) full digital image correlation (DIC) of a perforated tensile specimen, (2) a reduced-data scenario with a one-dimensional stretch profile and global force-displacement curve, and (3) a heterogeneous matrix-inclusion system in which both phases constitutive laws are identified and generalized to twenty-two previously unseen samples.