Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
Abstract
We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.
An interpretable identification framework based on universal neural ordinary differential equations (UNODEs), symbolic regression, and parameter refinement is developed that is competitive on autonomous polynomial systems and more effective in recovering compact symbolic structures for non-polynomial and explicitly time-varying dynamics.
Qing-Tong Dong· Engineering Research Express· 0 citations
Neural ordinary differential equations (ODEs) are widely used in neuroscience to model the collective activity of neurons during behavioral tasks. The high dimensionality of their parameter and activity spaces, however, often make it challenging to infer and interpret the fundamental features of their dynamics. In this study, we employ recent nonlinear dynamical system techniques to uncover the core dynamics of several Neural ODEs used in contemporary neuroscience. Specifically, using a data-driven approach, we identify Spectral Submanifolds (SSMs), i.e., low-dimensional attracting invariant manifolds tangent to the eigenspaces of fixed points. The internal dynamics of SSMs serve as nonlinear models that reduce the dimensionality of the full RNNs by orders of magnitude. Through low-dimensional, SSM-reduced models, we give mathematically precise definitions of line and ring attractors, which are intuitive concepts commonly used to explain decision-making and working memory. This unprecedented level of understanding of Neural ODEs obtained from SSM reduction enables the interpretation of mathematically well-defined and robust structures in neuronal dynamics, leading to predictions about the neural computations underlying behavior. Spectral submanifolds can be used to reduce recurrent neural networks to low-dimensional models, revealing their core dynamics. Here, authors uncover robust structures underlying decision-making and working-memory tasks, providing predictions about the underlying behavior of neural computations.
A. Marraffa, R. Krause, V. Mante et al.· Nature Communications· 0 citations
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.
This thesis advances the training and scalability of NCDEs through three complementary contributions, building on neural rough differential equations, which reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.
An eigenanalysis framework is developed that reveals the dynamical origin of inference-time error growth and introduces a stability-promoting loss that explicitly regularizes Jacobian-driven error amplification, improving both forecast accuracy and dynamical robustness.
C. Ainslie, P. Hassanzadeh, Michael W. Mahoney et al.· 0 citations
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
