Accurately predicting the long-term behavior of chaotic systems is crucial for various applications such as climate modeling. However, achieving such predictions typically requires iterative computations over a dense spatiotemporal grid to account for the unstable nature of chaotic systems, which is expensive and impractical in many real-world situations. An alternative approach to such a full-resolved simulation is using a coarse grid and then correcting its errors through a \textit{closure model}, which approximates the overall information from fine scales not captured in the coarse-grid simulation. Recently, ML approaches have been used for closure modeling, but they typically require a large number of training samples from expensive fully-resolved simulations (FRS). In this work, we prove an even more fundamental limitation, i.e., the standard approach to learning closure models suffers from a large approximation error for generic problems, no matter how large the model is, and it stems from the non-uniqueness of the mapping. We propose an alternative end-to-end learning approach using a physics-informed neural operator (PINO) that overcomes this limitation by not using a closure model or a coarse-grid solver. We first train the PINO model on data from a coarse-grid solver and then fine-tune it with (a small amount of) FRS and physics-based losses on a fine grid. The discretization-free nature of neural operators means that they do not suffer from the restriction of a coarse grid that closure models face, and they can provably approximate the long-term statistics of chaotic systems. In our experiments, our PINO model achieves a 120x speedup compared to FRS with a relative error $\sim 5\%$. In contrast, the closure model coupled with a coarse-grid solver is $58$x slower than PINO while having a much higher error $\sim205\%$ when the closure model is trained on the same FRS dataset.

翻译：准确预测混沌系统的长期行为对于气候建模等应用至关重要。然而，实现此类预测通常需要在密集的时空网格上进行迭代计算，以考虑混沌系统的不稳定性，这在许多实际场景中既昂贵又不切实际。替代这种全分辨率模拟的方法是使用粗网格，然后通过\textit{闭合模型}校正其误差，该模型近似粗网格模拟中未捕获的细尺度整体信息。最近，机器学习方法已被用于闭合建模，但它们通常需要从昂贵的全分辨率模拟（FRS）中获取大量训练样本。在这项工作中，我们证明了一个更为根本的局限性，即学习闭合模型的标准方法对于一般性问题存在较大的近似误差，无论模型规模多大，这源于映射的非唯一性。我们提出了一种替代的端到端学习方法，使用物理信息神经算子（PINO），该方法通过不使用闭合模型或粗网格求解器来克服这一限制。我们首先在粗网格求解器的数据上训练PINO模型，然后使用（少量）FRS数据和基于物理的损失函数在细网格上进行微调。神经算子的无离散化特性意味着它们不会受到闭合模型所面临的粗网格限制，并且可证明能够近似混沌系统的长期统计特性。在我们的实验中，与FRS相比，我们的PINO模型实现了120倍的加速，相对误差约为$5\%$。相比之下，当闭合模型在相同的FRS数据集上训练时，结合粗网格求解器的闭合模型比PINO慢58倍，同时具有更高的误差（约$205\%$）。