Learning differentiable solvers for systems with hard constraints
This addresses the challenge of incorporating physical constraints into neural network models for scientific computing applications, representing a novel method rather than an incremental improvement.
The authors tackled the problem of enforcing partial differential equation (PDE) constraints in neural networks by developing a differentiable PDE-constrained layer that can be integrated into any architecture, resulting in much lower test error compared to unconstrained training.
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our method leverages differentiable optimization and the implicit function theorem to effectively enforce physical constraints. Inspired by dictionary learning, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that accurately satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.