🤖 AI Summary
To address the challenges of training unsupervised neural PDE solvers under scarce data and the limitations of traditional discretization methods (e.g., finite differences) in spatiotemporal resolution and computational cost, this paper proposes a particle-ensemble modeling framework grounded in a probabilistic representation of PDEs: the PDE solution is interpreted as the evolution of a stochastic particle system, eliminating explicit grid-based discretization. Innovatively integrating Monte Carlo sampling with neural networks, we introduce the first method that jointly couples Heun’s method (for advection) and neighborhood-based PDF expectation estimation (for diffusion) along particle trajectories. The approach is inherently robust to strong spatiotemporal variations and enables stable training with coarse time steps. Evaluated on convection–diffusion, Allen–Cahn, and Navier–Stokes equations, it achieves up to a 3.2× accuracy improvement and over 40% higher computational efficiency compared to state-of-the-art unsupervised baselines.
📝 Abstract
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.
