Inferring solutions of differential equations using noisy multi-fidelity data

被引:207
作者
Raissi, Maziar [1 ]
Perdikaris, Paris [2 ]
Karniadakis, George Em [1 ]
机构
[1] Brown Univ, Div Appl Math, Providence, RI 02912 USA
[2] MIT, Dept Mech Engn, Cambridge, MA 02139 USA
关键词
Machine learning; Integro-differential equations; Multi-fidelity modeling; Uncertainty quantification;
D O I
10.1016/j.jcp.2017.01.060
中图分类号
TP39 [计算机的应用];
学科分类号
081203 ; 0835 ;
摘要
For more than two centuries, solutions of differential equations have been obtained either analytically or numerically based on typically well-behaved forcing and boundary conditions for well-posed problems. We are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations. We develop data-driven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-differential operators. The only observables are scarce noisy multi-fidelity data for the forcing and solution that are not required to reside on the domain boundary. The resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning. This general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time-integration, and is scalable to high-dimensions. (C) 2017 Elsevier Inc. All rights reserved.
引用
收藏
页码:736 / 746
页数:11
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