This work empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data and 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.
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.
This manuscript gives an overview of the filtering mechanism behind the weak formulation and provides a bias-variance error decomposition, and combines a weak formulation with a kernel learning strategy to propose Weak-form Kernel Ridge Regression (WKRR) for learning dynamical systems.
Max Kreider, John Harlim, Daning Huang· 2 citations· ⚡1
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.
It is demonstrated that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.
It is shown that the method consistently identifies compact governing equations while maintaining strong long-horizon predictive accuracy across canonical nonlinear systems and representative industrial processes, even under noisy and distribution-shifted data.
We introduce a data-driven framework that maps noisy oscillatory time series directly onto the Hopf normal form, enabling inference of underlying dynamics without knowledge of governing equations. By embedding the normal form in a probabilistic state-space model, the method jointly infers latent states and system parameters, yielding robust estimates of the natural frequency, Floquet exponent, and asymptotic phase even far from the bifurcation point and under strong noise. Combined with complex Gaussian process regression, the approach further reconstructs phase and amplitude sensitivity functions from data. Benchmarks on the van der Pol oscillator demonstrate substantially improved accuracy and noise robustness compared with existing phase-based and regression methods. This work establishes a direct bridge between normal-form theory and statistical inference, providing a general and practical route to low-dimensional descriptions of oscillatory dynamics in complex systems.
This study investigates the capability of deep neural networks to infer the time-evolution of the Rössler system a canonical chaotic oscillator by leveraging initial conditions and forcing parameters as input variables and underscores the need for hybrid approaches to address long-term instability.
A. Fateh, Harrag Abdelmalek, F. Mohamed et al.· International Journal of App...· 0 citations
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