mlsanityData to 7 Oct 2026

Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation

arXiv 2302.05104 ↗First posted 10 Feb 2023cs.LG cs.NA math.NAcited by 11

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Abstract (arXiv)

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.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 48 of its 111 references that are in the map. First posted on arXiv 10 Feb 2023.

Builds on

48 of 111 references.

References · 48

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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 6; 2026: 4.

11 papers.

  1. 2025PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systemscited by 5
  2. 2024Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equationscited by 2
  3. 2025PIMRL: Physics-Informed Multi-Scale Recurrent Learning for Burst-Sampled Spatiotemporal Dynamicscited by 1
  4. 2025Spectral-inspired Operator Learning with Limited Data and Unknown Physicscited by 1
  5. 2025Nowcast3D: Reliable precipitation nowcasting via gray-box learningcited by 1
  6. 2025OmniFluids: Physics Pre-trained Modeling of Fluid Dynamicscited by 0
  7. 2025Stable spectral neural operator for learning stiff PDE systems from limited datacited by 0
  8. 2026Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogatescited by 0
  9. 2026PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamicscited by 0
  10. 2026NPSolver: Neural Poisson Solver with Iterative Physics Supervisioncited by 0
  11. 2026Neural Harmonic Measure Operatorcited by 0