PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning, explaining why these methods can be data-efficient. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of 3D elliptic PDEs from input-output data and achieves an exponential convergence rate with respect to the size of the training dataset with an exceptionally high probability of success.
翻译:PDE学习是一个新兴领域,将物理和机器学习结合起来,从实验数据中恢复未知的物理系统。虽然深层次学习模式传统上需要大量培训数据,但最近的PDE学习技术在有限的数据可用性下取得了惊人的成果。不过,这些成果是经验性的。我们的工作为PDE学习所需的投入-产出培训对数提供了理论保障,解释了这些方法为什么能够提高数据效率。具体地说,我们利用随机数字数字线性代数和PDE理论,得出一种可被证实的数据效率算法,从输入-产出数据中回收3D 椭圆式 PDE的解决方案操作员,并在培训数据集的规模方面实现指数趋同率,成功率极高。</s>