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This research introduces a "Weak Penalty" Neural ODE framework designed to improve the forecasting accuracy of chaotic dynamical systems under noisy observational conditions. By utilizing weak formulations instead of standard L2-loss, the method effectively filters noise while preserving long-term invariant properties in high-dimensional models like climate datasets. The approach offers a robust alternative for training neural differential equations where traditional methods struggle with instability and error amplification.
A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series
arXiv:2511.06609v5 Announce Type: replace Abstract: The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noisy observations of deterministic dynamics, which severely degrade the performance of data-driven models. In chaotic dynamical systems, where small initial errors amplify exponentially, it is particularly difficult to develop a model from noisy data that achieves short-term accuracy while preserving long-term invariant properties. To overcome this, we consider the weak formulation as a complementary approach to the classical L2-loss function for training models of dynamical systems. We empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data. This insight explains why a weak form loss function is analogous to fitting a model to filtered data and provides a practical way to parameterize the weak form. Subsequently, we demonstrate how this approach overcomes the instability and inaccuracy of standard Neural ODE (NODE) in modeling chaotic systems. Through numerical examples, we show that our proposed training strategy, the Weak Penalty NODE, is computationally efficient, solver-agnostic, and yields accurate and robust forecasts across benchmark chaotic systems and a real-world climate dataset.
Abstract:The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noisy observations of deterministic dynamics, which severely degrade the performance of data-driven models. In chaotic dynamical systems, where small initial errors amplify exponentially, it is particularly difficult to develop a model from noisy data that achieves short-term accuracy while preserving long-term invariant properties. To overcome this, we consider the weak formulation as a complementary approach to the classical L2-loss function for training models of dynamical systems. We empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data. This insight explains why a weak form loss function is analogous to fitting a model to filtered data and provides a practical way to parameterize the weak form. Subsequently, we demonstrate how this approach overcomes the instability and inaccuracy of standard Neural ODE (NODE) in modeling chaotic systems. Through numerical examples, we show that our proposed training strategy, the Weak Penalty NODE, is computationally efficient, solver-agnostic, and yields accurate and robust forecasts across benchmark chaotic systems and a real-world climate dataset.
Machine Learning (cs.LG); Dynamical Systems (math.DS)
arXiv:2511.06609 [cs.LG]
(or arXiv:2511.06609v5 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2511.06609 Focus to learn more arXiv-issued DOI via DataCite
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