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Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space
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Proposes Holistic Physics Mixer (HPM), which couples point-wise physical states with fixed Laplace-Beltrami spectral basis via a learned softmax modulation to unify spectral and attention neural operators for PDE solving.
Problem
Neural operators for PDEs face a dichotomy: spectral methods generalize well with limited data and across resolutions but lack point-level flexibility, while attention methods are flexible but overfit with scarce data and generalize poorly to unseen resolutions.
Approach
Holistic Physics Transform projects features onto spectral basis functions modulated pointwise by a softmax of an MLP of the physical state (H(x,Φ)=Softmax(MLP(x))⊙Φ), so the mixer is both a spectral operator and a linear-attention-like integral operator; the authors claim this unifies both families as special cases.
Key results
- On structured-mesh benchmarks HPM reports relative L2 of 4.59e-3 on Darcy Flow vs 5.70e-3 for Transolver and 7.34e-2 on Navier-Stokes vs 9.00e-2 for Transolver.
- On unstructured meshes HPM achieves the lowest error on all five problems, e.g., 1.84e-4 on Heat Transfer vs 2.70e-4 for NORM.
- Zero-shot Airfoil generalization: HPM 1.57e-2 at higher resolution vs Transolver 6.95e-2 in the additional experiment table.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Transolver with HPM | Transolver · Darcy Flow, Airfoil, Navier-Stokes, and Plasticity | Relative L2 Error: Darcy Flow: 5.70e-3\std{1.00e-4} -> 4.59e-3\std{1.94e-4}; Airfoil: 5.30e-3\std{1.00e-4} -> 4.72e-3\std{1.51e-4}; Navier- | Helped |
| replace SpecSolver with HPM | SpecSolver · Irregular Darcy, Pipe Turbulence, Heat Transfer, Composite, and Blood Flow | Relative L2 Error: Irregular Darcy: 7.96e-3\std{7.19e-5} -> 7.38e-3\std{6.20e-5}; Pipe Turbulence: 1.11e-2\std{1.00e-3} -> 8.26e-3\std{7.60 | Helped |
| use point-wise Softmax coupling rather than the alternative coupling forms (ablation) | HPM · Airfoil | Relative Error: Form 1: 4.38e-3; Form 2: 4.86e-3; Form 3: 5.47e-3; Form 4: 4.69e-3; Form 5: 5.24e-3 | Helped |
| replace point-wise coupling with global-pooled coupling (ablation) | HPM · Airfoil | Relative Error: 4.38e-3 -> 5.47e-3 | Hurt |
| replace Softmax coupling with Sigmoid coupling (ablation) | HPM · Airfoil | Relative Error: 4.38e-3 -> 4.86e-3 | Hurt |
| replace multiplicative coupling with additive coupling (ablation) | HPM · Airfoil | Relative Error: 4.38e-3 -> 4.69e-3 | Hurt |
| concatenate physical states and spectral basis before coupling (ablation) | HPM · Airfoil | Relative Error: 4.38e-3 -> 5.24e-3 | Hurt |
| increase head count from 1 to multiple heads (ablation) | HPM · Darcy Flow | Relative Error: 1 head: 4.77e-3; 4 heads: 4.56e-3; 8 heads: 4.38e-3; 16 heads: 4.54e-3 | Helped |
| reduce depth from 8 to 4 layers (ablation) | HPM · Darcy Flow | L2 Error: 4.38e-3 -> 4.71e-3 | Hurt |
| add high-frequency basis in middle layers instead of early layers (ablation) | SpecSolver · Airfoil | Relative Error: 5.12e-3 -> 4.84e-3 | Helped |
| use HPM rather than SpecSolver | SpecSolver · Darcy Flow, 200–1000 training trajectories; Navier-Stokes, 200–1000 training trajectories | Relative L2 Error: Darcy Flow: 200: 1.10e-2 -> 1.06e-2; 400: 7.32e-3 -> 6.66e-3; 600: 6.20e-3 -> 6.03e-3; 800: 5.64e-3 -> 4.98e-3; 1000: 5. | Helped |
| use HPM rather than SpecSolver | SpecSolver · Darcy Flow and Navier-Stokes at frequency numbers 16, 32, 64, and 128 | Relative L2 Error: Darcy Flow: 16: 1.04e-2 -> 5.10e-3; 32: 8.08e-3 -> 5.02e-3; 64: 6.15e-3 -> 4.85e-3; 128: 5.31e-3 -> 4.38e-3. Navier-Stok | Helped |
Surprising findings
- Fixed spectral basis enhancements guided by HPM's learned modulation patterns (adding high frequencies in middle layers) improved SpecSolver from 5.33e-3 to 4.84e-3 on Darcy Flow.
Limitations
- The coupling function is a single simple design; more sophisticated couplings are not explored.
- Evaluation limited to PDE benchmarks; applications to time-series or vision are left for future work.
- Universal approximation claim relies on a Monte-Carlo approximation requiring dense sampling and a continuous bounded coupling function.
Experimental setup
Datasets: Darcy Flow Navier-Stokes Airfoil Plasticity Irregular Darcy Pipe Turbulence Heat Transfer Composite Blood Flow
Benchmarks: Darcy Flow Airfoil Navier-Stokes Plasticity Irregular Darcy Pipe Turbulence Heat Transfer Composite Blood Flow
Metrics: relative L2 error inference time parameter count
Baselines: FNO WMT U-FNO Geo-FNO U-NO F-FNO LSM NORM SpecSolver Galerkin HT-Net OFORMER
Builds on: Fourier neural operator Transolver Geo-FNO Galerkin transformer Laplace-Beltrami eigenfunctions
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.88)
- Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning (NeurIPS 2024 · similarity 0.87)
- Ultra Fast PDE Solving via Physics Guided Few-step Diffusion (NeurIPS 2026 · similarity 0.87)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.87)
- Mosaic: A Benchmark Suite for Differentiable Physics Solvers (NeurIPS 2026 · similarity 0.87)
- Active Learning for Neural PDE Solvers (ICLR 2025 · similarity 0.86)
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