PhD Comprehensive Seminar | Phuong Dong Le, Neural Network Architecture to Solve the Partial Differential Equations

MC 6460

Candidate

Phuong Dong Le | Applied Mathematics, University of Waterloo

Title

Neural Network Architecture to Solve the Partial Differential Equations

Abstract

Neural network models have shown a great potential in solving partial differential equations (PDE). Once trained with numerical simulation data, these models can provide faster alternative to traditional simulators and be efficient. However, they suffer from the generalization problem. There have been previous works that address the issue by applying universal approximation theorem for operator (DeepONet) using two sub-neural-networks. This approach has been generalized to neural network models that can learn mappings between function spaces. A recent work of neural operator (FNO) has been formulated as a new method by parameterizing the integral kernel into a Fourier space. We perform experiments on inviscid Burgers' equation and Darcy flow on proposed neural network operator architectures. Our preliminary results showed that they can achieve an error rate up to 0.0024 on Darcy's flow.