Research map / Probabilistic and scientific modeling / Uncertainty and Gaussian processes
FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators
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Introduces FNOPE, a simulation-based inference method that uses Fourier Neural Operators with a flow-matching objective and a GP base distribution to estimate posteriors over function-valued parameters at arbitrary discretizations.
Problem
Simulation-based inference works poorly for function-valued (spatially/temporally varying) parameters, which are usually inferred on fixed discretizations or via basis expansions that scale badly and need many simulations.
Approach
Combine a Fourier Neural Operator backbone with flow-matching posterior estimation, use a Gaussian process base distribution whose lengthscale matches the retained Fourier modes, and use a non-uniform Fourier transform plus masking and positional-noise augmentation so posteriors can be evaluated on unseen discretizations and jointly with vector-valued parameters.
Key results
- On the 1000-dimensional linear Gaussian task, FNOPE reaches near-zero SWD with 10^2 simulations, whereas NPE and FMPE (spectral) need up to 10^5.
- On Darcy flow (~16k parameters), FNOPE has much higher normalized posterior log-probability than FMPE (raw) and produces sharper posteriors than spectral baselines.
- On the Antarctic ice shelf task, FNOPE at 10^2 simulations matches NPE (raw) performance that requires about 10^4 simulations, and infers all 500 parameters rather than 50.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FNO blocks for posterior estimation instead of spectral-coefficient NPE/FMPE | FNOPE · Linear Gaussian | SWD | Helped |
| Use FNOPE rather than spectral NPE/FMPE | Darcy flow · Darcy flow | predictive MSE | Helped |
| Use FNOPE rather than FMPE (raw) | Darcy flow · Darcy flow | posterior log-probability | Helped |
| Use FNOPE rather than Simformer | SIRD · SIRD | predictive MSE | No clear effect |
| Use FNOPE rather than other methods at low simulation budgets | Antarctic ice · Antarctic ice test simulations | predictive MSE | Helped |
| Use FNOPE rather than NPE (raw) | Antarctic ice real data · Antarctic ice | predictive performance | Helped |
| Use 20 rather than 40 observation time points | FNOPE on SIRD · SIRD | inference performance | Hurt |
| Add sufficiently many Fourier modes to the FNO blocks (ablation) | FNOPE · Linear Gaussian | SWD | Helped |
| Change other tested hyperparameters (ablation) | FNOPE · Linear Gaussian | SWD | No clear effect |
| Use the lengthscale heuristic for the GP base noise (ablation) | FNOPE · Darcy flow | overall performance | Helped |
| Use FNOPE with data augmentation rather than FNOPE (fix) (ablation) | FNOPE · Linear Gaussian | SWD | Hurt |
Surprising findings
- Independent Gaussian white noise base distributions are unsuitable; a GP base distribution with a lengthscale tied to the retained Fourier modes was needed.
- FNOPE performance drops somewhat when only 20 observation time points are used for the SIRD task, reflecting the need for enough points for the Fourier transform.
Limitations
- FNO assumes limited high-frequency content, so parameters with discontinuities are poorly suited.
- Requires sufficiently many observation points for (approximate) Fourier transforms, so very sparse observations are not supported.
- Computational cost still scales exponentially with domain dimension.
Experimental setup
Datasets: Gaussian linear simulator (synthetic) SIRD epidemiological simulator (synthetic) Darcy flow simulator (synthetic) Antarctic ice shelf radar internal reflection horizons simulator and real radar observation
Benchmarks: Gaussian linear simulator SIRD Darcy flow Antarctic ice shelf mass balance
Metrics: Sliced-Wasserstein distance predictive MSE SBC error of diagonal posterior log-probability per pixel
Baselines: NPE (spectral) FMPE (spectral) FMPE (raw) Simformer NPE (raw) iFNO
Builds on: flow matching posterior estimation neural posterior estimation Fourier Neural Operator rectified flow Simformer
Closest papers by meaning
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