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FUSE: Fast Unified Simulation and Estimation for PDEs
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Proposes FUSE, a unified framework that pairs a Fourier Neural Operator forward surrogate with a flow-matching posterior estimator over finite-dimensional PDE parameters, so parameter inference and continuous field prediction are trained and evaluated together.
Problem
Joint prediction of continuous PDE fields and statistical inference of the discrete parameters that generate them; existing approaches address these separately or use restrictive VAE-based joint models.
Approach
Decompose the forward-inverse error via the triangle inequality into a measure-matching term (flow-matching posterior estimation over parameters given functional inputs) and an operator-learning term (FNO with a band-limited lifting from parameters to functions), trained with decoupled losses and combined at evaluation for propagated uncertainty.
Key results
- On ACB, FUSE with estimated parameters reaches CRPS 1.84 and relative L1 error 0.58% vs. 4.47 (UNet), 6.65 (InVAErt), and 19.84 (cVANO) CRPS.
- On PWP at information level 1, FUSE achieves relative L1 error 0.86% and CRPS 1.31, versus 3.79 CRPS for UNet and 11.62 for InVAErt.
- On out-of-distribution ACB samples, FUSE has the lowest errors (relative L1 1.63% with estimated parameters vs. 3.43% for UNet and 15.35% for cVANO); cVANO outperforms FUSE on the inverse problem only for PWP with missing information.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace FNO layers with UNet layers and omit Fourier transforms (ablation) | FUSE · PWP, Level 1 | CRPS: 3.79 \pm 0.97 -> 1.31 $\pm$ 0.69 | Helped |
| replace FNO layers with UNet layers and omit Fourier transforms (ablation) | FUSE · PWP, Level 1 | relative L1 error: 3.81 $\pm$ 2.03\% -> 0.86 $\pm$ 0.93} \% | Helped |
| replace FNO layers with UNet layers and omit Fourier transforms (ablation) | FUSE · ACB, estimated parameters | CRPS: 4.47 $\pm$ 2.67 -> 1.84 $\pm$ 1.63 | Helped |
| replace FNO layers with UNet layers and omit Fourier transforms (ablation) | FUSE · ACB, estimated parameters | relative L1 error: 2.13 $\pm$ 1.16 \% -> 0.58 $\pm$ 0.59} \% | Helped |
| use FMPE optimal-transport paths instead of diffusion-based paths (ablation) | conditional posterior estimator · ACB | CRPS: $4.96 \pm 2.23$ -> $1.84 \pm 1.63$ | Helped |
| increase posterior sample count from 128 to 8192 (ablation) | FMPE · ACB | CRPS: $1.84 \pm 1.63$ -> $1.44 \pm 1.64$ | Helped |
| increase training data from 128 to 8192 samples (ablation) | FUSE · ACB | CRPS: $16.10 \pm 4.53$ -> $1.84 \pm 1.63$ | Helped |
| train without masking, using only fingertip PPG as input (ablation) | FUSE · PWP, Level 3 | relative L1 error: 3.6\%, roughly half the error of the masked model | Helped |
| add the additional output-through-model training losses (ablation) | FUSE · experiments | evaluation performance | No clear effect |
| provide less input information (ablation) | FUSE · PWP, Levels 1–3 | CRPS: 1.31 $\pm$ 0.69 -> 3.18 $\pm$ 1.40 -> 8.48 \pm 2.55 | Hurt |
Surprising findings
- Training with decoupled losses and combining at evaluation outperformed end-to-end backpropagation through the full network.
- cVANO outperformed FUSE on the inverse problem for the PWP case with missing input information.
Limitations
- Assumes the given PDE parameterization is a perfect model; structural model uncertainty is not considered.
- Does not handle measurement errors in continuous inputs; no Bayesian treatment of input noise.
- Not yet resolution-invariant in the latent space; parameters are finite-dimensional only.
Experimental setup
Datasets: pulse wave propagation reduced-order model dataset (published by prior work) PyCLES atmospheric cold bubble simulations
Benchmarks: pulse wave propagation (PWP) atmospheric cold bubble (ACB)
Metrics: CRPS relative L1 error relative L2 error
Baselines: cVANO InVAErt GAROM UNet ablation
Builds on: Fourier Neural Operator Flow-Matching Posterior Estimation InVAErt cVANO
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