Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids

Scientific computing using deep learning has seen significant advancements in recent years. There has been growing interest in models that learn the operator from the parameters of a partial differential equation (PDE) to the corresponding solutions. Deep Operator Network (DeepONet) and Fourier Neural operator, among other models, have been designed with structures suitable for handling functions as inputs and outputs, enabling real-time predictions as surrogate models for solution operators. There has also been significant progress in the research on surrogate models based on graph neural networks (GNNs), specifically targeting the dynamics in time-dependent PDEs. In this paper, we propose GraphDeepONet, an autoregressive model based on GNNs, to effectively adapt DeepONet, which is well-known for successful operator learning. GraphDeepONet exhibits robust accuracy in predicting solutions compared to existing GNN-based PDE solver models. It maintains consistent performance even on irregular grids, leveraging the advantages inherited from DeepONet and enabling predictions on arbitrary grids. Additionally, unlike traditional DeepONet and its variants, GraphDeepONet enables time extrapolation for time-dependent PDE solutions. We also provide theoretical analysis of the universal approximation capability of GraphDeepONet in approximating continuous operators across arbitrary time intervals.

HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork

Fast and accurate predictions for complex physical dynamics are a significant challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. This study proposes HyperDeepONet, which uses the expressive power of the hypernetwork to enable the learning of a complex operator with a smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a particular case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet successfully learned various operators with fewer computational resources compared to other benchmarks.

AL-PINNs: Augmented Lagrangian relaxation method for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) has become a prominent application of deep learning in scientific computation, as it is a powerful approximator of solutions to nonlinear partial differential equations (PDEs). There have been numerous attempts to facilitate the training process of PINNs by adjusting the weight of each component of the loss function, called adaptive loss balancing algorithms. In this paper, we propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs). We treat the initial and boundary conditions as constraints for the optimization problem of the PDE residual. By employing Augmented Lagrangian relaxation, the constrained optimization problem becomes a sequential max-min problem so that the learnable parameters $\lambda$'s adaptively balance each loss component. Our theoretical analysis reveals that the sequence of minimizers of the proposed loss functions converges to an actual solution for the Helmholtz, viscous Burgers, and Klein--Gordon equations. We demonstrate through various numerical experiments that AL-PINNs yields a much smaller relative error compared with that of state-of-the-art adaptive loss balancing algorithms.