Inferring solutions of dierential equations using noisy multi-delity data
Inferring solutions of dierential equations using noisy multi-delity data
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发表时间：
2016
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通讯作者：
M. Raissi;P. Perdikaris;G. Karniadakis
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作者：
M. Raissi;P. Perdikaris;G. Karniadakis
两个多世纪以来，对于适定问题，微分方程解无论是解析的还是数值的，都是基于典型的良态强迫条件和边界条件得到的。我们正在通过在概率机器学习和微分方程式之间建立一个接口，从根本上改变这种范式。我们开发了一般线性方程的数据驱动算法，该算法使用为相应的积分方向算子量身定制的高斯过程先验。唯一可观测的是不需要驻留在区域边界上的、用于强迫和解的稀少的噪声多重解数据。由此得到的预测后验分布量化了不确定性，并自然地导致了通过主动学习的自适应解再现。这种通用框架绕过了数值离散化的暴政以及时间积分的一致性和稳定性问题，并且可以扩展到高维。
For more than two centuries, solutions of dierential 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 dierential equations. We develop datadriven algorithms for general linear equations using Gaussian process priors tailored to the corresponding integro-dierenti al operators. The only observables are scarce noisy multi-delity 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 renement 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.