Wansheng Wang

Wansheng Wang, Qingguo Hong

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size $H$ and the fine grid size $h$ satisfy $H=O(h^{\frac{r-1}{r}})$. Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.

Wansheng Wang, Jie Wang, Jinping Li et al.

We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. In this novel algorithm, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on deep FBSDE method. To derive the error estimates for this deep learning algorithm, the convergence of Markovian iteration, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated. Two numerical examples are provided to show the efficiency of this proposed algorithm.

Zaijun Ye, Chen-Song Zhang, Wansheng Wang

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between training and inference, leading to compounding errors in long-term autoregressive predictions. To address this issue, we propose Recurrent Neural Operators (RNOs)-a novel framework that integrates recurrent training into neural operator architectures. Instead of conditioning each training step on ground-truth inputs, RNOs recursively apply the operator to their own predictions over a temporal window, effectively simulating inference-time dynamics during training. This alignment mitigates exposure bias and enhances robustness to error accumulation. Theoretically, we show that recurrent training can reduce the worst-case exponential error growth typical of teacher forcing to linear growth. Empirically, we demonstrate that recurrently trained Multigrid Neural Operators significantly outperform their teacher-forced counterparts in long-term accuracy and stability on standard benchmarks. Our results underscore the importance of aligning training with inference dynamics for robust temporal generalization in neural operator learning.