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Abstract The fast simulation of dynamical systems is a key challenge in many scientific and engineering applications, such as weather forecasting, disease control, and drug discovery. With the recent success of deep learning, there is increasing interest in using neural networks to solve differential equations in a data‐driven manner. However, existing methods are either limited to specific types of differential equations or require large amounts of data for training. This restricts their practicality in many real‐world applications, where data is often scarce or expensive to obtain. To address this, a novel multi‐modal foundation model, namedFMint(FoundationModel based onInitialization) is proposed, to bridge the gap between human‐designed and data‐driven models for the fast simulation of dynamical systems. Built on a decoder‐only transformer architecture with in‐context learning, FMint utilizes both numerical and textual data to learn a universal error correction scheme for dynamical systems, using prompted sequences of coarse solutions from traditional solvers. The model is pre‐trained on a corpus of 400K ordinary differential equations (ODEs), and extensive experiments are performed on challenging ODEs that exhibit chaotic behavior and of high dimensionality. The results demonstrate the effectiveness of the proposed model in terms of both accuracy and efficiency compared to classical numerical solvers, highlighting FMint's potential as a general‐purpose solver for dynamical systems. This approach achieves an accuracy improvement of 1 to 2 orders of magnitude over state‐of‐the‐art dynamical system simulators, and delivers a 5X speedup compared to traditional numerical algorithms. The code for FMint is available athttps://github.com/margotyjx/FMint. 

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.