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Neural networks have been identified as powerful tools for the study ofcomplex systems. A noteworthy example is the neural network differentialequation (NN DE) solver, which can provide functional approximations to thesolutions of a wide variety of differential equations. Such solvers producerobust functional expressions, are well suited for further manipulations on thequantities of interest (for example, taking derivatives), and capable ofleveraging the modern advances in parallelization and computing power. However,there is a lack of work on the role precise error quantification can play intheir predictions: usually, the focus is on ambiguous and/or global measures ofperformance like the loss function and/or obtaining global bounds on the errorsassociated with the predictions. Precise, local error quantification is seldompossible without external means or outright knowledge of the true solution. Weaddress these concerns in the context of dynamical system NN DE solvers,leveraging learnt information within the NN DE solvers to develop methods thatallow them to be more accurate and efficient, while still pursuing anunsupervised approach that does not rely on external tools or data. We achievethis via methods that can precisely estimate NN DE solver prediction errorspoint-wise, thus allowing the user the capacity for efficient and targetederror correction. We exemplify the utility of our methods by testing them on anonlinear and a chaotic system each.

Local error quantification for Neural Network Differential Equation solvers