Source code for the training pipeline, tasks, and models used in this work, is available as part of the supplementary material. We used the same Adam [48] optimizer for all our experiments and a learning rate of 0.001, and a batch size of 128. For solving the differential equations both during ground truth data generation as well as with the neural ODEs, we use the Tsitouras 5/4 Runge-Kutta (Tsit5) method from DifferentialEquations.jl [36]. A.1 Coupled Pendulum The coupled pendulum dynamics are defined as We train the MP-NODE on a dataset of 500 trajectories, each randomly initialized with state values between [ π/2, π/2] for the θ and [ 1, 1] for θ, with a time step of 0.1s and each trajectory 10s long. The dataset is normalized through Z-score normalization.

In this study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations (ODEs). While traditional numerical methods a re effective for many ODEs, they often struggle to achieve convergence in problems involving high stiffness, shocks, irregular domains, singular perturbations, high dimensions, or boundary discontinuities. Alternatively, PINNs offer a powerful approach for handling challenging numerical scenarios. In this study, classical ODE problems are employed as controlled testbeds to systematically evaluate the accuracy, training efficiency, and generalization capability under controlled conditions of the PINNs framework. Although not a universal solution, PINNs can achieve superior results by embedding physical laws directly into the learning process. We first analyze the existence and uniqueness properties of several benchmark problems and subsequently validate the PINNs methodology on these model systems. Our results demonstrate that for complex problems to converge to correct solutions, the loss function components data loss, initial condition loss, and residual loss must be appropriately balanced through careful weighting. We further establish that systematic tuning of hyperparameters, including network depth, layer width, activation functions, learning rate, optimization algorithms, w eight initialization schemes, and collocation point sampling, plays a crucial role in achieving accurate solutions. Additionally, embedding prior knowledge and imposing hard constraints on the network architecture, without loss the generality of the ODE system, significantly enhances the predictive capability of PINNs.

Learning governing equations from data is central to understanding the behavior of physical systems across diverse scientific disciplines, including physics, biology, and engineering. The Sindy algorithm has proven effective in leveraging sparsity to identify concise models of nonlinear dynamical systems. In this paper, we extend sparsity-driven approaches to real-time learning by integrating a cornerstone algorithm from control theory -- the Kalman filter (KF). The resulting Sindy Kalman Filter (SKF) unifies both frameworks by treating unknown system parameters as state variables, enabling real-time inference of complex, time-varying nonlinear models unattainable by either method alone. Furthermore, SKF enhances KF parameter identification strategies, particularly via look-ahead error, significantly simplifying the estimation of sparsity levels, variance parameters, and switching instants. We validate SKF on a chaotic Lorenz system with drifting or switching parameters and demonstrate its effectiveness in the real-time identification of a sparse nonlinear aircraft model built from real flight data.

Data-driven discovery of model equations is a powerful approach for understanding the behavior of dynamical systems in many scientific fields. In particular, the ability to learn mathematical models from data would benefit systems biology, where the complex nature of these systems often makes a bottom up approach to modeling unfeasible. In recent years, sparse estimation techniques have gained prominence in system identification, primarily using parametric paradigms to efficiently capture system dynamics with minimal model complexity. In particular, the Sindy algorithm has successfully used sparsity to estimate nonlinear systems by extracting from a library of functions only a few key terms needed to capture the dynamics of these systems. However, parametric models often fall short in accurately representing certain nonlinearities inherent in complex systems. To address this limitation, we introduce a novel framework that integrates sparse parametric estimation with nonparametric techniques. It captures nonlinearities that Sindy cannot describe without requiring a priori information about their functional form. That is, without expanding the library of functions to include the one that is trying to be discovered. We illustrate our approach on several examples related to estimation of complex biological phenomena.

The nonlinear nature of chaotic systems results in extreme sensitivity to initial conditions and highly intricate dynamical behaviors, posing fundamental challenges for accurately predicting their evolution. To overcome the limitation that conventional approaches fail to capture both local features and global dependencies in chaotic time series simultaneously, this study proposes a parallel predictive framework integrating Transformer and Bidirectional Long Short-Term Memory (BiLSTM) networks. The hybrid model employs a dual-branch architecture, where the Transformer branch mainly captures long-range dependencies while the BiLSTM branch focuses on extracting local temporal features. The complementary representations from the two branches are fused in a dedicated feature-fusion layer to enhance predictive accuracy. As illustrating examples, the model's performance is systematically evaluated on two representative tasks in the Lorenz system. The first is autonomous evolution prediction, in which the model recursively extrapolates system trajectories from the time-delay embeddings of the state vector to evaluate long-term tracking accuracy and stability. The second is inference of unmeasured variable, where the model reconstructs the unobserved states from the time-delay embeddings of partial observations to assess its state-completion capability. The results consistently indicate that the proposed hybrid framework outperforms both single-branch architectures across tasks, demonstrating its robustness and effectiveness in chaotic system prediction.

Scientific data, from cellular snapshots in biology to celestial distributions in cosmology, often consists of static patterns from underlying dynamical systems. These snapshots, while lacking temporal ordering, implicitly encode the processes that preserve them. This work investigates how strongly such a distribution constrains its underlying dynamics and how to recover them. We introduce the Equilibrium flow method, a framework that learns continuous dynamics that preserve a given pattern distribution. Our method successfully identifies plausible dynamics for 2-D systems and recovers the signature chaotic behavior of the Lorenz attractor. For high-dimensional Turing patterns from the Gray-Scott model, we develop an efficient, training-free variant that achieves high fidelity to the ground truth, validated both quantitatively and qualitatively. Our analysis reveals the solution space is constrained not only by the data but also by the learning model's inductive biases. This capability extends beyond recovering known systems, enabling a new paradigm of inverse design for Artificial Life. By specifying a target pattern distribution, we can discover the local interaction rules that preserve it, leading to the spontaneous emergence of complex behaviors, such as life-like flocking, attraction, and repulsion patterns, from simple, user-defined snapshots.

Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Dated: September 5, 2025) Finding the governing equations from data by sparse optimization has become a popular approach to deterministic modeling of dynamical systems. Considering the physical situations where the data can be imperfect due to disturbances and measurement errors, we show that for many chaotic systems, widely used sparse-optimization methods for discovering governing equations produce models that depend sensitively on the measurement procedure, yet all such models generate virtually identical chaotic attractors, leading to a striking limitation that challenges the conventional notion of equation-based modeling in complex dynamical systems. Calculating the Koopman spectra, we find that the different sets of equations agree in their large eigenvalues and the differences begin to appear when the eigenvalues are smaller than an equation-dependent threshold. The results suggest that finding the governing equations of the system and attempting to interpret them physically may lead to misleading conclusions. It would be more useful to work directly with the available data using, e.g., machine-learning methods. In physical science, a methodology of biblical importance is developing a quantitative model by extracting a set of governing equations from experimental data.

This paper investigates the generalisability of Koopman-based representations for chaotic dynamical systems, focusing on their transferability across prediction and control tasks. Using the Lorenz system as a testbed, we propose a three-stage methodology: learning Koopman embeddings through autoencoding, pre-training a transformer on next-state prediction, and fine-tuning for safety-critical control. Our results show that Koopman embeddings outperform both standard and physics-informed PCA baselines, achieving accurate and data-efficient performance. Notably, fixing the pre-trained transformer weights during fine-tuning leads to no performance degradation, indicating that the learned representations capture reusable dynamical structure rather than task-specific patterns. These findings support the use of Koopman embeddings as a foundation for multi-task learning in physics-informed machine learning. A project page is available at https://kikisprdx.github.io/.

Remark 4. The result of Theorem 2 also holds for unstable orbits Remark 5. F or RNNs with ReLU activation functions there are finite compartments in the phase Assume for the sake of contradiction that k W k 1 . There is a direct link between the norms of the Jacobians of the RNN along trajectories and the EVGP . Note that this system is non-autonomous, that is externally forced due to the r.h.s. of eqn. Rössler system Another prime textbook example for a chaotic system is the Rössler system [76] This strongly suggests that the dynamics governing the time series are chaotic. All datasets used were standardized (i.e., centered with unit variance) prior to training.

Recent advances in artificial intelligence have highlighted the remarkable capabilities of neural network (NN)-powered systems on classical computers. However, these systems face significant computational challenges that limit scalability and efficiency. Quantum computers hold the potential to overcome these limitations and increase processing power beyond classical systems. Despite this, integrating quantum computing with NNs remains largely unrealized due to challenges posed by noise, decoherence, and high error rates in current quantum hardware. Here, we propose a novel quantum echo-state network (QESN) design and implementation algorithm that can operate within the presence of noise on current IBM hardware. We apply classical control-theoretic response analysis to characterize the QESN, emphasizing its rich nonlinear dynamics and memory, as well as its ability to be fine-tuned with sparsity and re-uploading blocks. We validate our approach through a comprehensive demonstration of QESNs functioning as quantum observers, applied in both high-fidelity simulations and hardware experiments utilizing data from a prototypical chaotic Lorenz system. Our results show that the QESN can predict long time-series with persistent memory, running over 100 times longer than the median T1 and T2 of the IBM Marrakesh QPU, achieving state-of-the-art time-series performance on superconducting hardware.