Research map / Probabilistic and scientific modeling / Differential equations and PDEs
A Neural-Preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes a lightweight multiresolution neural preconditioner with spatially varying convolution kernels, used inside a steepest-descent-with-orthogonalization solver, to solve mixed Dirichlet/Neumann Poisson systems for fluid simulation.
Problem
Solving discrete Poisson equations with mixed Dirichlet and Neumann boundary conditions on changing domains (e.g., free-surface fluid simulation), where multigrid and other preconditioners must be rebuilt at each step and prior neural preconditioners fail with Dirichlet conditions.
Approach
A linear multi-resolution (multigrid-like) network whose convolution kernels are spatially varying functions of a geometry/boundary-condition image, trained unsupervised with a residual-norm loss and used as a preconditioner in NPSDO (preconditioned steepest descent with orthogonalization), which handles nonsymmetric, nonlinear-free preconditioners.
Key results
- Ours is fastest on 95.6% of 2,400 benchmark systems, accounting for 98.0% of total solve time versus unpreconditioned CG.
- On average over benchmark suites, the proposed solver takes 17.2 iterations and 0.267 s versus AMGCL at 11.7 iterations and 1.691 s and CG at 623.1 iterations and 3.232 s.
- The network has about 25k parameters with 4 coarsening levels, and uses 1.5 GiB GPU RAM at 128^3 versus 5.2 GiB for FluidNet and 8.5 GiB for DCDM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the neural-preconditioned solver instead of AmgX | NPSDO (Ours) · All test suites (average) | runtime: 0.819 -> 0.267 | Helped |
| Use the neural-preconditioned solver instead of CG | NPSDO (Ours) · FluidNet examples | runtime: 0.5615 -> 0.05237 | Helped |
| Use the neural-preconditioned solver instead of DCDM | NPSDO (Ours) · FluidNet examples | runtime: 12.35 -> 0.05237 | Helped |
| Use AmgX instead of NPSDO | NPSDO (Ours) · Waterflow torus 128³ | runtime: 0.048 -> 0.046 | Hurt |
| Use AmgX instead of NPSDO | NPSDO (Ours) · Waterflow torus 256³ | runtime: 0.338 -> 0.346 | Hurt |
| Replace spatially varying kernels with a standard CNN and masked output (ablation) | NPSDO (Ours) · CNN masked-output comparison | runtime: 0.027 -> 0.1025 | Hurt |
| Replace spatially varying kernels with a standard CNN and masked output (ablation) | NPSDO (Ours) · CNN masked-output comparison | residual: 4.61×10⁻⁷ -> 3.23×10⁻² | Hurt |
| Use the standard CNN with masked output instead of CG (ablation) | Standard CNN · CNN masked-output comparison | residual: 9.72 ×10⁻⁷ -> 3.23×10⁻² | Hurt |
| Enforce symmetry and positive definiteness by concatenating the network with its transpose (ablation) | Neural preconditioner · Symmetry analysis | solver runtime | Hurt |
| Enforce symmetry and positive definiteness by tying pre- and post-smoothing blocks as transposes (ablation) | Neural preconditioner · Symmetry analysis | solver runtime | Hurt |
| Use FluidNet as a standalone solver instead of NPSDO | FluidNet · FluidNet examples | final residual: 8.77 × 10⁻⁷ -> 2.035 | Hurt |
| Use FluidNet as a PSDO preconditioner instead of NPSDO | FluidNet PSDO · FluidNet examples | final residual: 8.77 × 10⁻⁷ -> 2.10 × 10⁻⁵ | Hurt |
Surprising findings
- DCDM fails to outperform CG and incomplete Cholesky in the authors' CUDA-accelerated setup, contrary to the original DCDM paper.
- Standard CNN with fixed kernels and masked output performed poorly, with residual increasing after the first few iterations.
Limitations
- Preconditioner is not symmetric or positive definite, and symmetric variants trained worse than CUDA-accelerated CG.
- Network cost does not scale with fluid occupancy, so it is less efficient on sparse fluid domains.
- Restricted to structured grids with constant-coefficient Poisson equations and scalar unknowns.
Experimental setup
Datasets: 107 Poisson matrices from 11 fluid simulation scenes, with Lanczos-Ritz-based right-hand sides
Benchmarks: twelve fluid simulation scenes at 128^3, 256x128x128, 256^3, 192^3, 384x128x128
Metrics: solve time iteration count speedup vs CG final residual peak GPU memory
Baselines: unpreconditioned CG AMGCL algebraic multigrid NVIDIA AmgX incomplete Cholesky CHOLMOD FluidNet DCDM
Builds on: DCDM FluidNet PCG UNet
Closest papers by meaning
- UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs (ICML 2024 · similarity 0.86)
- Neural operators meet conjugate gradients: The FCG-NO method for efficient PDE solving (ICML 2024 · similarity 0.85)
- BENO: Boundary-embedded Neural Operators for Elliptic PDEs (ICLR 2024 · similarity 0.85)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.85)
- Towards General Neural Surrogate Solvers with Specialized Neural Accelerators (ICML 2024 · similarity 0.85)
- Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUs (NeurIPS 2025 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.