Symbolic Neural ODEs: Learning interpretable models from time-series data

Chaotic time series are notoriously difficult to forecast. Small uncertainties in initial conditions amplify rapidly, while strong nonlinearities and regime dependent variability constrain predictability. Although modern deep learning often delivers strong short horizon accuracy, its black box nature limits scientific insight and practical trust in settings where understanding the underlying dynamics matters. To address this gap, we propose two complementary symbolic forecasters that learn explicit, interpretable algebraic equations from chaotic time series data. Symbolic Neural Forecaster (SyNF) adapts a neural network based equation learning architecture to the forecasting setting, enabling fully differentiable discovery of compact and interpretable algebraic relations. The Symbolic Tree Forecaster (SyTF) builds on evolutionary symbolic regression to search directly over equation structures under a principled accuracy complexity trade off. We evaluate both approaches in a rolling window nowcasting setting with one step ahead forecasting using several accuracy metrics and compare against a broad suite of baselines spanning classical statistical models, tree ensembles, and modern deep learning architectures. Numerical experiments cover a benchmark of 132 low dimensional chaotic attractors and two real world chaotic time series, namely weekly dengue incidence in San Juan and the Nino 3.4 sea surface temperature index. Across datasets, symbolic forecasters achieve competitive one step ahead accuracy while providing transparent equations that reveal salient aspects of the underlying dynamics.

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

Neural autoregressive models have rapidly emerged as powerful emulators of high-dimensional chaotic systems, yet their long-term instability and error growth remain poorly understood, leading to ad-hoc solutions. Here, we develop an eigenanalysis framework that reveals the dynamical origin of this error growth. By analyzing the Jacobian of the learned one-step update map with respect to the state, we show how inference-time error growth, and thus model stability, is governed by its spectral radius. Direct-step architectures (models that predict the next state from the previous one) generically admit unstable eigenvalues with magnitudes exceeding one, explaining the rapid divergence of these widely used models. In contrast, integration-constrained models (where the time derivative is estimated and integrated with a higher-order integrator) collapse their eigenspectrum onto the unit circle, yielding neutral stability and a universal linear error-scaling law. The largest eigenvalue of this Jacobian provides an architecture-agnostic, a priori diagnostic of short-term skill, long-term stability, and spectral bias, without requiring an expensive rollout. Leveraging this theory, we introduce a stability-promoting loss that explicitly regularizes Jacobian-driven error amplification, improving both forecast accuracy and dynamical robustness. Demonstrated across $29$ models spanning two architectures, several explicit and implicit integrators, and multiple loss functions on the Kuramoto-Sivashinsky system, our results establish a theoretical foundation for the design and evaluation of neural emulators of chaotic multi-scale dynamics. More broadly, our framework is a step toward the kind of a priori stability analysis that numerical analysis provides for discretizations of differential equations and that scientific machine learning currently lacks.