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Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes Structured Spectral Propagator (SSP), an analysis-propagation-synthesis neural PDE surrogate that evolves retained latent Fourier modes with a frequency-conditioned linear backbone plus a nonlinear spectral closure, evaluated on NS, SW and RD rollouts.

Problem

Autoregressive neural operator rollouts for time-dependent PDEs accumulate errors and drift over long horizons, and latent propagators tie reconstruction and propagation together so the update lacks structure.

Approach

Separate spatial-basis encoding from a compact propagation state, then advance retained latent Fourier modes with a shared temporal matrix modulated by a frequency-dependent gate, plus a near-zero-initialized nonlinear closure, integrated as residual Euler substeps with normality and orthogonality regularizers.

Key results

• On Navier-Stokes, SSP reaches relative L2 of 5.50E-2 versus 7.57E-2 for the best baseline F-FNO, a reduction of about 27%.
• On Reaction-Diffusion, SSP reaches L2 of 7.46E-3 versus 1.46E-2 for U-FNO, roughly a 49% reduction, consistent with the 'up to 48.9%' claim.
• On Shallow-Water temporal extrapolation (steps 51-90), SSP achieves the best E_max (2.55E-1) and BRMS (5.46E-4), while F-FNO has the lowest L2 (4.36E-2 vs SSP 4.49E-2).

Surprising findings

• UNO diverges to extremely large errors (about 1e11) during Shallow-Water extrapolation, and FNO, LNO and U-FNO also show severe error growth beyond the training horizon.
• On Navier-Stokes, KNO attains a lower E_max than SSP despite much worse L2 and spectral errors.

Limitations

• Results reported as mean over three seeds only with std in appendix; the gains are reported on three benchmarks only.
• Some baselines (LNO on RD) failed to train, and the paper does not detail compute or hyperparameter tuning fairness.

Experimental setup

Datasets: Navier-Stokes Shallow-Water Reaction-Diffusion

Benchmarks: Navier-Stokes Shallow-Water Reaction-Diffusion

Metrics: relative L2 error E_max f_low f_mid f_high f_mse BRMS

Baselines: FNO U-FNO F-FNO UNO KNO LNO CALM

Builds on: Fourier neural operator Koopman operator theory latent neural operator

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