NeuralStagger: accelerating physics-constrained neural PDE solver with spatial-temporal decomposition

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional $10\sim100\times$ speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

Monte Carlo Neural Operator for Learning PDEs via Probabilistic Representation

Neural operators, which use deep neural networks to approximate the solution mappings of partial differential equation (PDE) systems, are emerging as a new paradigm for PDE simulation. The neural operators could be trained in supervised or unsupervised ways, i.e., by using the generated data or the PDE information. The unsupervised training approach is essential when data generation is costly or the data is less qualified (e.g., insufficient and noisy). However, its performance and efficiency have plenty of room for improvement. To this end, we design a new loss function based on the Feynman-Kac formula and call the developed neural operator Monte-Carlo Neural Operator (MCNO), which can allow larger temporal steps and efficiently handle fractional diffusion operators. Our analyses show that MCNO has advantages in handling complex spatial conditions and larger temporal steps compared with other unsupervised methods. Furthermore, MCNO is more robust with the perturbation raised by the numerical scheme and operator approximation. Numerical experiments on the diffusion equation and Navier-Stokes equation show significant accuracy improvement compared with other unsupervised baselines, especially for the vibrated initial condition and long-time simulation settings.

LordNet: Learning to Solve Parametric Partial Differential Equations without Simulated Data

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data which can be costly to collect, resulting in a chicken-egg dilemma and limiting its usage in solving PDEs. To jump out of the dilemma, we propose a general data-free paradigm where the neural network directly learns physics from the mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss and identify the challenge that the neural network must have the capacity to model the long range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. Therefore, we propose the low-rank decomposition network (LordNet) which is tunable and also efficient to model various entanglements. Specifically, LordNet learns a low-rank approximation to the global entanglements with simple fully connected layers, which extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and Navier-Stokes equation demonstrate that the physical constraints by the MSR loss can lead to better accuracy and generalization ability of the neural network. In addition, LordNet outperforms other modern neural network architectures in both PDEs with the fewest parameters and the fastest inference speed. For Navier-Stokes equation, the learned operator is over 50 times faster than the finite difference solution with the same computational resources.

Learning Physics-Informed Neural Networks without Stacked Back-propagation

Physics-Informed Neural Network (PINN) has become a commonly used machine learning approach to solve partial differential equations (PDE). But, facing high-dimensional second-order PDE problems, PINN will suffer from severe scalability issues since its loss includes second-order derivatives, the computational cost of which will grow along with the dimension during stacked back-propagation. In this paper, we develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. We further discuss the model capacity and provide variance reduction methods to address key limitations in the derivative estimation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.

Cooperative Policy Learning with Pre-trained Heterogeneous Observation Representations

Multi-agent reinforcement learning (MARL) has been increasingly explored to learn the cooperative policy towards maximizing a certain global reward. Many existing studies take advantage of graph neural networks (GNN) in MARL to propagate critical collaborative information over the interaction graph, built upon inter-connected agents. Nevertheless, the vanilla GNN approach yields substantial defects in dealing with complex real-world scenarios since the generic message passing mechanism is ineffective between heterogeneous vertices and, moreover, simple message aggregation functions are incapable of accurately modeling the combinational interactions from multiple neighbors. While adopting complex GNN models with more informative message passing and aggregation mechanisms can obviously benefit heterogeneous vertex representations and cooperative policy learning, it could, on the other hand, increase the training difficulty of MARL and demand more intense and direct reward signals compared to the original global reward. To address these challenges, we propose a new cooperative learning framework with pre-trained heterogeneous observation representations. Particularly, we employ an encoder-decoder based graph attention to learn the intricate interactions and heterogeneous representations that can be more easily leveraged by MARL. Moreover, we design a pre-training with local actor-critic algorithm to ease the difficulty in cooperative policy learning. Extensive experiments over real-world scenarios demonstrate that our new approach can significantly outperform existing MARL baselines as well as operational research solutions that are widely-used in industry.