Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.
Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds
Physikalisch-Technische Bundesanstalt (PTB)（德国联邦物理技术研究院）；Technical University of Berlin（柏林工业大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）；Eastern Mediterranean University（东地中海大学）
Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes
Department of Mathematics, Indian Institute of Science, Bangalore（印度科学研究所数学系，班加罗尔）；Department of Computing and Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学科学系）
Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks
Global Center for Science and Engineering（全球科学与工程中心）；Waseda University（早稻田大学）；Faculty of Engineering（工学院）；Tokyo University of Agriculture and Technology（东京农业大学）
The University of Tokyo（东京大学）；Mitsubishi Electric Corporation（三菱电机公司）
In this article, two types of methods from different perspectives based on spectral normalization are described for ensuring the stability of the system controlled by a neural network. The first one is that the L2 gain of the feedback system is bounded less than 1 to satisfy the stability condition derived from the small-gain theorem. While explicitly including the stability condition, the first method may provide an insufficient performance on the neural network controller due to its strict stability condition. To overcome this difficulty, the second one is proposed, which improves the perfor…
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online…
Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds
Centre for Mathematical Sciences, University of Cambridge, UK（剑桥大学数学科学中心，英国）；Department of Mathematics, Imperial College London, UK（伦敦帝国理工学院数学系，英国）；Department of Mechanical Engineering, University of Washington, USA（美国华盛顿大学机械工程系）
