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Physics-Informed Neural Networks (PINNs) have gained much attention invarious fields of engineering thanks to their capability of incorporatingphysical laws into the models. PINNs integrate the physical constraints byminimizing the partial differential equations (PDEs) residuals on a set ofcollocation points. The distribution of these collocation points appears tohave a huge impact on the performance of PINNs and the assessment of thesampling methods for these points is still an active topic. In this paper, wepropose a Fixed-Budget Online Adaptive Learning (FBOAL) method, whichdecomposes the domain into sub-domains, for training collocation points basedon local maxima and local minima of the PDEs residuals. The effectiveness ofFBOAL is demonstrated for non-parameterized and parameterized problems. Thecomparison with other adaptive sampling methods is also illustrated. Thenumerical results demonstrate important gains in terms of the accuracy andcomputational cost of PINNs with FBOAL over the classical PINNs withnon-adaptive collocation points. We also apply FBOAL in a complex industrialapplication involving coupling between mechanical and thermal fields. We showthat FBOAL is able to identify the high-gradient locations and even give betterpredictions for some physical fields than the classical PINNs with collocationpoints sampled on a pre-adapted finite element mesh built thanks to numericalexpert knowledge. From the present study, it is expected that the use of FBOALwill help to improve the conventional numerical solver in the construction ofthe mesh.

Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inference