Automatically Polyconvex Strain Energy Functions using Neural Ordinary Differential Equations

Data-driven methods are becoming an essential part of computational mechanics due to their unique advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form approximations. However, imposing the physics-based mathematical requirements that any material model must comply with is not straightforward for data-driven approaches. In this study, we use a novel class of neural networks, known as neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy function with respect to the deformation gradient, a condition needed for the existence of minimizers for boundary value problems in elasticity. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy function with respect to the invariants of the right Cauchy-Green deformation tensor. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations. The framework is general and can be used to model a large class of materials. Here we focus on hyperelasticity, but polyconvex strain energies are a core building block for other problems in elasticity such as viscous and plastic deformations. We therefore expect our methodology to further enable data-driven methods in computational mechanics