Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Fourier Neural Operator for Parametric Partial Differential Equations
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Parameterize the convolution kernel directly in Fourier space using truncated modes | Neural operator · Burgers' equation, Darcy Flow, Navier-Stokes | relative error: 30% lower on Burgers' Equation, 60% lower on Darcy Flow, and 30% lower on Navier Stokes | Helped |
| Use the FNO-3D space-time architecture instead of FNO-2D with recurrent time propagation | FNO · Navier-Stokes, ν=1e-3, T=50, N=1000 | relative error: 0.0128 -> 0.0086 | Helped |
| Use the FNO-3D space-time architecture instead of FNO-2D with recurrent time propagation | FNO · Navier-Stokes, ν=1e-4, T=30, N=10000 | relative error: 0.0834 -> 0.0820 | Helped |
| Use the FNO-3D space-time architecture instead of FNO-2D with recurrent time propagation | FNO · Navier-Stokes, ν=1e-4, T=30, N=1000 | relative error: 0.1559 -> 0.1918 | Hurt |
| Use the FNO-3D space-time architecture instead of FNO-2D with recurrent time propagation | FNO · Navier-Stokes, ν=1e-5, T=20, N=1000 | relative error: 0.1556 -> 0.1893 | Hurt |
| Replace direct Fourier-mode parameterization with a linear parameterization of R (ablation) | FNO · PDE experiments | performance | No clear effect |
| Replace direct Fourier-mode parameterization with a neural-network parameterization of R (ablation) | FNO · PDE experiments | performance | Hurt |
| Train on lower-resolution data and evaluate at higher resolution | FNO-3D · Navier-Stokes super-resolution, 64×64×20 to 256×256×80 | zero-shot super-resolution capability | Helped |
| Use only k_max,j=12 parameterized modes while retaining nonlinear activations | FNO · Navier-Stokes, ν=1e-3 | relative error: ≤ 1% | Helped |
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