A natural-gradient optimizer that expands neural networks when their tangent space misaligns with the ideal Hilbert-space gradient reaches target accuracy on several toy regression, PDE, and model-reduction problems.
A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations
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abstract
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.
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Expansive Natural Neural Gradient Flows for Energy Minimization
A natural-gradient optimizer that expands neural networks when their tangent space misaligns with the ideal Hilbert-space gradient reaches target accuracy on several toy regression, PDE, and model-reduction problems.