| Abstract |
Neural networks have become increasingly effective for approximating solutions to partial differential equations (PDEs). This talk presents three advances that improve both accuracy and computational efficiency. First, I introduce an augmented Lagrangian formulation of the physics-informed loss that strengthens constraint enforcement and improves accuracy near domain boundaries. Second, I develop efficient architectures based on hypernetworks and graph neural networks that learn PDE solution operators with markedly small model sizes. Finally, I describe Neural-Galerkin schemes with low rank approximations for operator learning, which achieves a favorable accuracy-efficiency trade-off. |
