Local error quantification for efficient neural network dynamical system solvers
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Neural Networks have been identified as potentially powerful tools for the study of complex systems. A noteworthy example is the Neural Network Differential Equation (NN DE) solver, which can provide functional approximations to the solutions of a wide variety of differential equations. However, there is a lack of work on the role precise error quantification can play in their predictions: most variants focus on ambiguous and/or global measures of performance like the loss function. We address this in the context of dynamical system NN DE solvers, leveraging their learnt information to develop more accurate and efficient solvers, while still pursuing a completely unsupervised, data free approach. We achieve this by providing methods for quantifying the performance of NN DE solvers at local scales, thus allowing the user the capacity for efficient and targeted error correction. We showcase the utility of these methods by testing them on a nonlinear and a chaotic system each.
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