Learning Nonlinear Constitutive Laws via Neural Network-Based Elastic Energy Modeling

We propose a differentiable framework for learning nonlinear constitutive laws in soft materials by directly modeling the elastic energy as a neural network. Using Discrete Differential Geometry (DDG)-based strain formulations, the approach captures complex, strain-dependent stiffness behaviors that cannot be represented by classical linear elasticity theory. The neural energy model is trained to reproduce observed equilibrium configurations, with gradients computed efficiently through equilibrium states using an adjoint-based sensitivity formulation. This approach enables self-supervised learning from static or quasi-static data without requiring explicit force measurements. Once trained, the model provides a compact, differentiable constitutive representation that generalizes across deformation regimes and material types. In addition, the framework can construct reduced-order surrogate models for rods, shells, and other soft robotic structures, enabling faster simulations and design iterations. By combining geometric mechanics with differentiable energy learning, this work aims to advance simulation-based investigation and design in soft robotics and soft matter physics.