Monte Carlo Neural PDE Solver for Learning PDEs Via Probabilistic Representation

In scenarios with limited available data, training the function-to-functionneural PDE solver in an unsupervised manner is essential. However, theefficiency and accuracy of existing methods are constrained by the propertiesof numerical algorithms, such as finite difference and pseudo-spectral methods,integrated during the training stage. These methods necessitate carefulspatiotemporal discretization to achieve reasonable accuracy, leading tosignificant computational challenges and inaccurate simulations, particularlyin cases with substantial spatiotemporal variations. To address theselimitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) fortraining unsupervised neural solvers via the PDEs' probabilisticrepresentation, which regards macroscopic phenomena as ensembles of randomparticles. Compared to other unsupervised methods, MCNP Solver naturallyinherits the advantages of the Monte Carlo method, which is robust againstspatiotemporal variations and can tolerate coarse step size. In simulating thetrajectories of particles, we employ Heun's method for the convection processand calculate the expectation via the probability density function ofneighbouring grid points during the diffusion process. These techniques enhanceaccuracy and circumvent the computational issues associated with Monte Carlosampling. Our numerical experiments on convection-diffusion, Allen-Cahn, andNavier-Stokes equations demonstrate significant improvements in accuracy andefficiency compared to other unsupervised baselines. The source code will bepublicly available at: https://github.com/optray/MCNP.