
Perturbation Theory for Quantum Information
We report lowestorder series expansions for primary matrix functions of...
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A higher order perturbation approach for electromagnetic scattering problems on random domains
We consider timeharmonic electromagnetic scattering problems on perfect...
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Weak Form Theoryguided Neural Network (TgNNwf) for Deep Learning of Subsurface Single and Twophase Flow
Deep neural networks (DNNs) are widely used as surrogate models in geoph...
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Learning Potentials of Quantum Systems using Deep Neural Networks
Machine Learning has wide applications in a broad range of subjects, inc...
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Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem
We present a new hybrid physicsbased machinelearning approach to reser...
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Critical initialization of wide and deep neural networks through partial Jacobians: general theory and applications to LayerNorm
Deep neural networks are notorious for defying theoretical treatment. Ho...
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A Neural Network Perturbation Theory Based on the Born Series
Deep Learning has become an attractive approach towards various databased problems of theoretical physics in the past decade. Its protagonists, the deep neural networks (DNNs), are capable of making accurate predictions for data of arbitrarily high complexity. A wellknown issue most DNNs share is their lack of interpretability. In order to explain their behavior and extract physical laws they have discovered during training, a suitable interpretation method has, therefore, to be applied posthoc. Due to its simplicity and ubiquity in quantum physics, we decide to present a rather general interpretation method in the context of twobody scattering: We find a onetoone correspondence between the n^thorder Born approximation and the n^thorder Taylor approximation of deep multilayer perceptrons (MLPs), that predict Swave scattering lengths a_0 for discretized, attractive potentials of finite range. This defines a perturbation theory for MLPs similarily to Born approximations defining a perturbation theory for a_0. In the case of shallow potentials, lowerorder approximations, that can be argued to be local interpretations of respective MLPs, reliably reproduce a_0. As deep MLPs are highly nested functions, the computation of higherorder partial derivatives, which is substantial for a Taylor approximation, is an effortful endeavour. By introducing quantities we refer to as propagators and vertices and that depend on the MLP's weights and biases, we establish a graphtheoretical approach towards partial derivatives and local interpretability. Similar to Feynman rules in quantum field theories, we find rules that systematically assign diagrams consisting of propagators and vertices to the corresponding order of the MLP perturbation theory.
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