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PDEfuncta: Spectrally-Aware Neural Representation for PDE Solution Modeling
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Proposes Global Fourier Modulation (GFM), which injects fixed Fourier bases into each layer of a modulated INR via weight reparameterization, and PDEfuncta, a meta-learned bidirectional latent representation of paired PDE solution fields.
Problem
Modulated implicit neural representations (Functa-style) suffer from spectral bias and struggle to represent high-frequency PDE solution fields across multiple instances with a shared network, and existing approaches do not support bidirectional forward/inverse inference.
Approach
Extends Fourier reparameterization of weights to the multi-instance modulated setting: a shared learnable base matrix plus a sample-specific latent-derived modulation added to the coefficients over a fixed multi-frequency Fourier basis, and jointly encodes input-output field pairs with one shared latent.
Key results
- GFM reaches 42.474 PSNR on convection vs 27.410 for Scale and 27.330 for FiLM in single-INR reconstruction (up to ~15 dB gain).
- On unseen convection coefficients with latent fitting, GFM attains 30.830 PSNR (Setting 2) while FiLM reaches 11.927.
- On Euler-NACA neural operator task, PDEfuncta achieves L2 error 0.0051, lower than CORAL (0.0059) and MARBLE (0.0058); on Pipe, Geo-FNO (0.0066) beats PDEfuncta (0.0081).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GFM instead of FiLM | SIREN modulation · Convection | PSNR: 27.330 -> 42.474 | Helped |
| Use GFM instead of FiLM | SIREN modulation · Helmholtz #1 | PSNR: 33.611 -> 39.139 | Helped |
| Use GFM instead of FiLM | SIREN modulation · Helmholtz #2 | PSNR: 22.771 -> 29.033 | Helped |
| Use GFM instead of FiLM | SIREN modulation · Navier-Stokes | PSNR: 31.904 -> 34.754 | Helped |
| Use GFM instead of FiLM | SIREN modulation · Kuramoto-Sivashinsky | PSNR: 22.297 -> 29.827 | Helped |
| Use GFM instead of Shift | PDEfuncta paired-field reconstruction · FWI (A) | PSNR: 25.879 -> 28.757 | Helped |
| Use GFM instead of FiLM | PDEfuncta paired-field reconstruction · FWI (U) | PSNR: 30.026 -> 32.761 | Helped |
| Use GFM instead of FiLM | PDEfuncta paired-field reconstruction · Navier-Stokes (A) | PSNR: 42.649 -> 43.166 | Helped |
| Use GFM instead of FiLM | PDEfuncta paired-field reconstruction · Navier-Stokes (U) | PSNR: 43.105 -> 43.792 | Helped |
| Use PDEfuncta instead of MARBLE | Euler-NACA forward operator · Euler-NACA | Test L2-error: $0.0058$ -> $0.0051$ | Helped |
| Use PDEfuncta instead of Geo-FNO | Pipe forward operator · Pipe | Test L2-error: $0.0066$ -> 0.0081 | Hurt |
| Use GFM instead of FiLM | SIREN modulation with latent interpolation · Convection, unseen coefficients Setting 1 | PSNR: 10.613 -> 32.586 | Helped |
Surprising findings
- On the Pipe dataset, PDEfuncta is not best; Geo-FNO achieves lower L2 error (0.0066 vs 0.0081).
- Shift, Scale and FiLM fail badly on unseen convection coefficients (PSNR around 8-13), while GFM generalizes.
Limitations
- Performance depends on careful tuning of the frequency basis hyperparameters (n_high, n_low, n_phase).
- Pipe neural operator result is not state-of-the-art; Geo-FNO outperforms PDEfuncta there.
- Theorem on gradient ratios holds only up to an error term epsilon and is borrowed from prior work.
Experimental setup
Datasets: 1D convection equation 2D Helmholtz equations 2D incompressible Navier-Stokes Kuramoto-Sivashinsky equation OpenFWI Euler-NACA airfoil Pipe flow
Benchmarks: 1D convection Helmholtz #1 Helmholtz #2 Navier-Stokes Kuramoto-Sivashinsky OpenFWI (FWI) Euler-NACA airfoil Pipe
Metrics: PSNR MSE L2 relative error
Baselines: Shift Scale FiLM Spatial Functa FNO UNet Geo-FNO FFNO CORAL MARBLE
Builds on: Functa Fourier reparameterization SIREN MAML FiLM
Closest papers by meaning
- Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains (ICML 2024 · similarity 0.86)
- Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows (ICLR 2025 · similarity 0.86)
- Maximum Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators (ICML 2025 · similarity 0.85)
- Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators (NeurIPS 2026 · similarity 0.85)
- Solving High Frequency and Multi-Scale PDEs with Gaussian Processes (ICLR 2024 · similarity 0.85)
- Amortized Fourier Neural Operators (NeurIPS 2024 · similarity 0.85)
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