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.

翻译：近年来，神经算子理论的最新进展使得快速、精确计算由偏微分方程描述的复杂系统解成为可能。尽管取得了巨大成功，当前基于神经算子的解决方案在处理长时间尺度的时空偏微分方程时仍面临重要挑战。具体而言，现有神经算子理论并未提供进行数据同化的系统化框架，也无法基于稀疏采样的含噪观测有效修正偏微分方程解随时间演化的过程。本文提出一种基于学习的状态空间方法，用于计算无限维半线性偏微分方程的解算子。通过利用半线性偏微分方程的结构特性以及函数空间中非线性观测器理论，我们开发了一种灵活的递归方法，该方法通过结合预测与修正操作，同时支持预测与数据同化。所提出的框架能够对长时间跨度实现快速精确的预测，处理非规则采样的含噪观测数据以修正解，并受益于该类偏微分方程空间与时间动力学的解耦特性。通过在Kuramoto-Sivashinsky方程、Navier-Stokes方程和Korteweg-de Vries方程上的实验表明，所提模型对噪声具有鲁棒性，且能以极小的计算开销利用任意数量的观测数据在长时间跨度上修正其预测结果。