Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation

TL;DR

This work addresses the lack of principled data assimilation for neural operator–based PDE solvers in long-time, noisy settings. It introduces NODA, a recursive neural operator framework that combines a prediction step with a correction step inspired by infinite-dimensional observer design to enable both forecasting and assimilation for semilinear PDEs. The method employs a Fourier Neural Operator predictor and learnable correction components, trained with a loss that balances state reconstruction and measurement reconstruction, and demonstrates robustness to noise on KS, NS, and KdV equations. Experiments show improved long-horizon predictions and effective data assimilation with modest computational overhead, highlighting practical impact for weather, remote sensing, and related dynamical systems applications.

Abstract

Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based solutions face important challenges when dealing with spatio-temporal PDEs over long time scales. Specifically, the current theory of NOs does not present a systematic framework to perform data assimilation and efficiently correct the evolution of PDE solutions over time based on sparsely sampled noisy measurements. In this paper, we propose a learning-based state-space approach to compute the solution operators to infinite-dimensional semilinear PDEs. Exploiting the structure of semilinear PDEs and the theory of nonlinear observers in function spaces, we develop a flexible recursive method that allows for both prediction and data assimilation by combining prediction and correction operations. The proposed framework is capable of producing fast and accurate predictions over long time horizons, dealing with irregularly sampled noisy measurements to correct the solution, and benefits from the decoupling between the spatial and temporal dynamics of this class of PDEs. We show through experiments on the Kuramoto-Sivashinsky, Navier-Stokes and Korteweg-de Vries equations that the proposed model is robust to noise and can leverage arbitrary amounts of measurements to correct its prediction over a long time horizon with little computational overhead.

Figures (8)

• Figure 1: Samples of a realization of a true trajectory of the Navier-Stokes equation for $t \in \{150,300,450\}$ (Top Row), and elementwise error plots for the corresponding predictions by NODA (Bottom Row).
• Figure 2: Prediction error for one realization of the KS equation. Warm-up with $t_H=40$ was performed. Plots depict the prediction period $(40,200]$ seconds. Ground truth evolution (Left). Error plots for NODA's solutions with $\alpha=0\%$ (Center) and $\alpha = 30\%$ (Right).
• Figure 3: Average RelMSE as a function of the warm-up length $t_H$ for the NS equation for different SNRs (Left); and average RelMSE as function of $t_f$ for different values of $\alpha$ for the KS equation, where a warm-up period of $t_H=40$ being used for NODA (Right).
• Figure 4: Average RelMSE as function of $t_f$ for different values of $\alpha$ for the KdV equation, where a warm-up period of $t_H=40$ being used for NODA.
• Figure 5: Sample of the predictionresults for the Korteweg-de Vries equation with $t_f=200$ seconds. Ground truth trajectory (Left Image). Solution estimated by NODA with $\alpha=0\%$ (Top left). Solution estimated by NODA with $\alpha=30\%$ (Top Right). Elementwise error plot for NODA's estimate without data assimilation (Bottom Left). Elementwise error plot for NODA's estimate with data assimilation (Bottom Right).