Asynchronous multimodal PINN pre-train framework based on TransVNet(MPP-TV) and its application in numerical solutions of the Cauchy problem for the Hamilton-Jacobi equation

The Hamilton-Jacobi(HJ) equation represents a class of highly nonlinear partial differential equations. Classical numerical techniques, such as finite element methods, face significant challenges when addressing the numerical solutions of such nonlinear HJ equations. However, recent advances in neural network-based approaches, particularly Physics-Informed Neural Networks (PINNs) and neural operator methods, have ushered in innovative paradigms for numerically solving HJ equations. In this work, we leverage the PINN approach, infused with the concept of neural operators. By encoding and extracting features from the discretized images of functions through TransVNet, which is a novel autoencoder architecture proposed in this paper, we seamlessly integrate Hamiltonian information into PINN training, thereby establishing a novel scientific computation framework. Additionally, we incorporate the vanishing viscosity method, introducing viscosity coefficients in our model, which equips our model to tackle potential singularities in nonlinear HJ equations. These attributes signify that our MPP-TV framework paves new avenues and insights for the generalized solutions of nonlinear HJ equations.