标题： 面向PDE基础模型的推理：一种奖励模型驱动的推理时缩放算法 显示英文标题

标题： Towards Reasoning for PDE Foundation Models: A Reward-Model-Driven Inference-Time-Scaling Algorithm

摘要： 偏微分方程（PDEs）是现代计算科学和工程的基础，且本质上计算成本高昂。 虽然基于PDE的基础模型在模拟这种复杂的时空现象方面表现出很大的潜力，但现有的模型仍然受到预训练数据集的限制，并且在自回归滚动性能方面存在困难，尤其是在分布外（OOD）情况下。 此外，它们对计算资源和训练数据有较大的需求，这阻碍了它们在许多关键应用中的使用。 受大型语言模型（LLMs）中“思考”策略的最新进展的启发，我们引入了第一个用于PDE的测试时计算（TTC）策略，该策略在推理过程中利用计算资源，以更少的训练样本和更小的模型实现更准确的预测。 我们通过两种类型的奖励模型来实现这一点，这些模型评估基于随机性的模型在时空一致性方面的预测。 我们在PDEGym基准的可压缩欧拉方程模拟上展示了这种方法，并表明与标准的非自适应自回归推理相比，TTC能够捕捉到改进的预测。 这个TTC框架标志着向更高级的推理算法或PDE建模迈出的基石性一步，包括构建基于强化学习的方法，可能会改变物理和工程中的计算流程。 显示英文摘要

摘要： Partial Differential Equations (PDEs) are the bedrock for modern computational sciences and engineering, and inherently computationally expensive. While PDE foundation models have shown much promise for simulating such complex spatio-temporal phenomena, existing models remain constrained by the pretraining datasets and struggle with auto-regressive rollout performance, especially in out-of-distribution (OOD) cases. Furthermore, they have significant compute and training data requirements which hamper their use in many critical applications. Inspired by recent advances in ``thinking" strategies used in large language models (LLMs), we introduce the first test-time computing (TTC) strategy for PDEs that utilizes computational resources during inference to achieve more accurate predictions with fewer training samples and smaller models. We accomplish this with two types of reward models that evaluate predictions of a stochastic based model for spatio-temporal consistency. We demonstrate this method on compressible Euler-equation simulations from the PDEGym benchmark and show that TTC captures improved predictions relative to standard non-adaptive auto-regressive inference. This TTC framework marks a foundational step towards more advanced reasoning algorithms or PDE modeling, inluding building reinforcement-learning-based approaches, potentially transforming computational workflows in physics and engineering.