LySep separates the layers and derivatives of a PINN into auxiliary variables, yielding a shallow, easier-to-optimize loss that remains provably consistent with the original PINN loss.
Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
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abstract
The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied mathematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems which we believe to be the most important topics for further study. This is not a complete overview over this quickly moving field, but we hope to provide a perspective which may be helpful especially to new researchers in the area.
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Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations
LySep separates the layers and derivatives of a PINN into auxiliary variables, yielding a shallow, easier-to-optimize loss that remains provably consistent with the original PINN loss.