Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation.

Area of Science:

• Computational Mathematics
• Machine Learning
• Scientific Computing

Background:

• Unsupervised training of neural PDE solvers is crucial with limited data.
• Existing methods face accuracy and efficiency constraints due to numerical algorithm properties like finite difference and pseudo-spectral methods.
• These methods require careful spatiotemporal discretization, causing computational challenges and inaccuracies with high variations.

Purpose of the Study:

• To propose the Monte Carlo Neural PDE Solver (MCNP Solver) for unsupervised neural solver training.
• To leverage the probabilistic representation of PDEs by modeling macroscopic phenomena as ensembles of random particles.
• To overcome the limitations of existing unsupervised methods in handling spatiotemporal variations.

Main Methods:

• The MCNP Solver utilizes a probabilistic approach, treating PDEs as ensembles of random particles.
• It incorporates Heun's method for simulating particle trajectories during convection.
• Expectation calculation during diffusion uses the probability density function of neighboring grid points.

Main Results:

• The MCNP Solver demonstrates robustness against spatiotemporal variations and tolerates coarse step sizes.
• Accuracy is enhanced by employing specific numerical techniques for convection and diffusion processes.
• Significant improvements in accuracy and efficiency were observed compared to other unsupervised baselines.

Conclusions:

• The MCNP Solver provides a more accurate and efficient approach for unsupervised neural PDE solving.
• Its probabilistic framework effectively handles complex spatiotemporal dynamics.
• The method shows promise for various PDE applications, including convection-diffusion, Allen-Cahn, and Navier-Stokes equations.