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Learning semilinear neural operators: A unified recursive framework for prediction and data assimilation.
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Proposes NODA, a recursive neural operator that alternates FNO-based prediction steps with learned Kalman-style correction steps derived from semilinear PDE observer theory, for long-horizon forecasting and data assimilation on KS, NS, and KdV equations.
Problem
Neural operators forecast spatio-temporal semilinear PDEs poorly over long horizons and lack a systematic way to assimilate sparse, noisy measurements.
Approach
Decompose the infinite-dimensional observer solution of semilinear PDEs into a prediction term (residual FNO step) and a correction term (learned observer gain with a learned measurement operator E), giving a recursive predict-update NO that handles missing or irregular measurements.
Key results
- On Navier-Stokes (Re=40, t_f=500, clean), NODA reports RelMSE x10^3 of 7 vs 18 for MNO and 136 for FNO.
- On Kuramoto-Sivashinsky (t_f=60, SNR 20dB), NODA achieves 123 x10^3 vs 144 for MNO and 298 for FNO, with MNO best in clean t_f=60 case (13).
- For NS (Re=40, SNR 30dB, t_f=1000), assimilating 30% of samples reduces RelMSE from 26 x10^3 (alpha=0%) to 9 x10^3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NODA instead of MNO for long-horizon prediction | KS · KS, t_f=120, SNR=20 dB | RelMSE (×10^3): 543$\pm16$ -> 493$\pm10$ | Helped |
| Use NODA instead of MNO for long-horizon prediction | KS · KS, t_f=120, SNR=30 dB | RelMSE (×10^3): 521$\pm16$ -> 483$\pm10$ | Helped |
| Use NODA instead of MNO for long-horizon prediction | KS · KS, t_f=120, SNR=∞ | RelMSE (×10^3): 410$\pm17$ -> 405$\pm12$ | Helped |
| Use NODA instead of MNO for long-horizon prediction | Navier-Stokes · Navier-Stokes, t_f=1000, SNR=20 dB | RelMSE (×10^3): 96$\pm12$ -> 32$\pm6$ | Helped |
| Use NODA instead of MNO for long-horizon prediction | Navier-Stokes · Navier-Stokes, t_f=1000, SNR=30 dB | RelMSE (×10^3): 87$\pm6$ -> 26$\pm4$ | Helped |
| Use NODA instead of MNO for long-horizon prediction | Navier-Stokes · Navier-Stokes, t_f=1000, SNR=∞ | RelMSE (×10^3): 26$\pm5$ -> 13$\pm2$ | Helped |
| Use NODA instead of MWNO for long-horizon prediction | KdV · KdV, t_f=200, SNR=20 dB | RelMSE (×10^3): 149 -> 102 | Helped |
| Use NODA instead of MWNO for long-horizon prediction | KdV · KdV, t_f=200, SNR=30 dB | RelMSE (×10^3): 114 -> 92 | Helped |
| Use NODA instead of MWNO for long-horizon prediction | KdV · KdV, t_f=200, SNR=∞ | RelMSE (×10^3): 92 -> 61 | Helped |
| Increase the assimilation sampling rate from α=0% to α=30% | NODA · Navier-Stokes, Re=40, t_f=1000, SNR=30 dB | RelMSE (×10^3): 26±4 -> 9±3 | Helped |
| Use the random measurement operator instead of the identity operator | NODA · Navier-Stokes, Re=40, α=0% | RelMSE (×10^3): 26 -> 42 | Hurt |
| Use the random measurement operator instead of the identity operator | NODA · Navier-Stokes, Re=40, α=30% | RelMSE (×10^3): 9 -> 25 | Hurt |
Surprising findings
- MNO outperforms NODA on the clean KS case with t_f=60 (13 vs 19 x10^3 at infinite SNR) in prediction-only mode.
Limitations
- Data assimilation experiments use a known or identity measurement operator C in most settings; random C is harder (RelMSE 42 vs 26 at alpha=0%).
- Prediction-only performance is not uniformly best; MNO wins some clean KS cases.
- Relies on a warm-up period with dense measurements (t_H=40 or 50) before assimilating sparse data.
Experimental setup
Datasets: Kuramoto-Sivashinsky simulated trajectories Navier-Stokes simulated trajectories (Re=40) Korteweg-de Vries simulated trajectories
Benchmarks: Kuramoto-Sivashinsky Navier-Stokes Korteweg-de Vries
Metrics: RelMSE
Baselines: FNO MNO MWNO Conv-LSTM LSTM C-LSTM GRU
Builds on: Fourier neural operator Markov neural operator infinite-dimensional observer design for semilinear PDEs
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