不直接预测状态，改预测变化率，提升神经PDE模型精度与灵活性。
Predicting Change, Not States: An Alternate Framework for Neural PDE Surrogates
- 改用预测时间导数+数值积分，替代直接预测下一状态
- 在精细离散化下显著提升模拟精度与稳定性
- 支持灵活时间步长，适合高分辨率数据训练
神经微分方程代理模型虽能快速模拟物理过程，但传统方法将时间演化视为黑箱，直接预测下一状态，过于简化且刚性。本文提出新框架：神经求解器预测时间导数，由常微分方程积分器推进解。该方法计算开销低，适用于多种模型结构和偏微分方程。实验表明，仅通过改变训练目标并引入数值积分，模型在精细离散化条件下显著提升精度与稳定性。此外，该框架摆脱特定时间离散化限制，支持推理时灵活调整时间步长，也适用于高分辨率训练数据。最后，我们分析了该框架的优势及其适用场景。
原文摘要 · Abstract (English)
Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a black box by directly predicting the next state. While this is a natural and easy framework for applying neural surrogates, it can be an over-simplified and rigid framework for predicting physics. In this work, we evaluate an alternate framework in which neural solvers predict the temporal derivative and an ODE integrator forwards the solution in time, which has little overhead and is broadly applicable across model architectures and PDEs. We find that by simply changing the training target and introducing numerical integration during inference, neural surrogates can gain accuracy and stability in finely-discretized regimes. Predicting temporal derivatives also allows models to not be constrained to a specific temporal discretization, allowing for flexible time-stepping during inference or training on higher-resolution PDE data. Lastly, we investigate why this framework can be beneficial and in what situations does it work well.
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