Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
arxiv(2023)
摘要
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.
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