Inducing Precision in Lagrangian Neural Networks : Proof of concept application on Chaotic systems
Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
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Keywords: Physics Informed Learning, Deep Learning, Neural Networks, Chaotic systems.
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TL;DR: Precision in Neural Networks can form functional solutions to chaotic systems.
Abstract: Solutions of dynamic systems that exhibit chaotic behavior are particularly sensitive to errors in initial/intermediate state estimates when long term dynamics is of interest. Lagrangian Neural Networks (LNN) are a class of physics induced learning methods that seamlessly integrate physical conservation laws into functional solutions, by forming a parametric Lagrangian for the system of interest. However it has been seen that the function approximation error associated with the parametric Lagrangian modelling could prove to be catastrophic for the prediction of long term dynamics of chaotic systems. This makes improving the precision of the parametric Lagrangian particularly crucial. Considering the same in this work a modified Lagrangian Neural Network approach is proposed, where a customized neural network architecture is designed to directly emphasize the relative importance of each significant bit in the Lagrangian estimates produced. We evaluate our method on two dynamic systems that are well known in the literature in exhibiting deterministic chaos, namely the double pendulum and Henon-Helies systems. Further, we compare the obtained solutions with those estimated by Finite Element solvers (under optimal conditions) to validate the relative accuracy. We observe that the trajectory deviations as a result of chaotic behavior can be significantly reduced by the process of explicitly enforcing the precision requirement for the parametric Lagrangian, as modelled using the proposed approach.
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Submission Number: 6604
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