Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
This work addresses computational challenges in PDE solving for domains like physics and engineering, offering a more efficient and accurate method, though it is incremental as it builds on existing unsupervised neural solver approaches.
The paper tackles the problem of training unsupervised neural PDE solvers under limited data by proposing the Monte Carlo Neural PDE Solver (MCNP Solver), which uses probabilistic representations to improve robustness against spatiotemporal variations and reduce computational costs, achieving significant improvements in accuracy and efficiency over existing baselines in experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations.
In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the trajectories of particles, we employ Heun's method for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines. The source code will be publicly available at: https://github.com/optray/MCNP.