Learning noise-induced transitions by multi-scaling reservoir computing

Noise-induced transitions, where noise plays a functional role in driving transitions between stable states, are prevalent in diverse natural systems. Examples include voltage and current state switches in circuits, genetic switches, biological homochirality, protein conformational transitions, and chemical reactions. Learning these transitions from time-series data, which often lacks underlying mathematical equations, presents a significant challenge. Conventional methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and recurrent neural networks (RNNs), typically focus on mitigating the effects of noise to extract deterministic dynamics, often failing to capture the essence of noise-induced phenomena. The slow transitions between stable states and the fast relaxation within them complicate the learning process. This paper addresses this challenge by proposing a novel approach using reservoir computing (RC), a machine learning technique known for its ability to learn complex dynamics from data.

Existing machine learning methods for learning dynamics from time series include SINDy, which aims to identify nonlinear dynamics and denoise data, and physics-informed neural networks, which can solve partial differential equations. However, these methods struggle to robustly handle large function libraries (SINDy due to non-convex optimization) or require extensive data (deep neural networks). Recurrent neural networks, although effective for learning dynamical systems, often fail to capture noise-induced transitions accurately. Previous work using RC for stochastic resonance or noise-induced transitions has either been limited or relied on unrealistic assumptions, such as pre-knowledge of deterministic dynamics. Therefore, a new model-free approach is needed that can learn noise-induced transitions solely from data.

The authors propose a multi-scaling reservoir computing framework. This approach utilizes the key hyperparameter *α* in the reservoir computing equation, which determines the timescale of the reservoir dynamics. By tuning *α*, the reservoir can be made to capture the slow timescale dynamics of the system, effectively separating it from the fast-scale noise. The reservoir computing architecture is given by the following equations: *r_t+1 = (1 − α)r_t + α tanh(Ar_t + W_inu_t)* *u_t+1 = W_outr_t+1* where *r* is the reservoir state vector, *u* is the state vector, *A* is the reservoir connection matrix, *W_in* is the input matrix, and *W_out* is the output matrix. The output matrix *W_out* is trained using linear regression to minimize the difference between the input and output time series. This training involves a regularization term to prevent overfitting. The training phase involves identifying an appropriate *α* to match the slow timescale dynamics and separate the noise distribution (*η*). The predicting phase utilizes the trained reservoir computer to simulate the slow-scale dynamics and then adds back the noise sampled from the separated noise distribution (for white noise) or learned from a second reservoir (for colored noise) via rolling prediction. The hyperparameters (*N*, *K_in*, *D*, *ρ*, *α*, *β*) are tuned using a combination of methods, including analysis of stable states and power spectral density (PSD) matching.

The proposed multi-scaling reservoir computing method accurately captures noise-induced transitions in various scenarios. For systems with white noise, it accurately predicts the statistics of transition time and the number of transitions. For systems with colored noise, it successfully predicts the specific transition time without needing pre-knowledge of the deterministic dynamics. The method was tested on a range of systems, including: * **1D bistable gradient system with white noise:** Accurately predicted average transition time and the number of transitions. * **2D bistable gradient and non-gradient systems with white noise:** Accurately predicted transition statistics, even with rotational dynamics introduced by non-detailed balance. * **1D bistable gradient system with colored noise (Lorenz noise):** Accurately predicted the specific transition time, demonstrating superiority over existing methods. * **2D tristable system with white noise:** Demonstrated applicability to multistable systems. * **Experimental protein folding data:** Accurately learned the statistics of transition times between folded states from a relatively small dataset (demonstrating that approximately 7500 timesteps were sufficient for accurate prediction), highlighting the method's efficiency and potential for reducing experimental data requirements.

The results demonstrate the effectiveness of the proposed multi-scaling reservoir computing method in learning noise-induced transitions in a model-free manner. The ability to accurately predict transition statistics and even specific transition times, especially for colored noise, marks a significant advance over existing methods. The success in analyzing experimental protein folding data further underscores the method's practical utility. The choice of hyperparameters, especially *α*, is crucial; larger *α* leads to faster timescales, while smaller *α* captures slower dynamics. This property allows the separation of noise from slow transitions. The hyperparameter search is guided by evaluating power spectral density (PSD) matches and stable state convergence. The method showed robustness within a range of *α* and *β* values. The finding that 7500 time steps were sufficient for protein folding predictions showcases the potential to reduce experimental data collection demands. The asymmetry of the potential wells did not present a major challenge, although using two sets of hyperparameters is an option for strongly asymmetric systems.

This study presents a general framework for learning noise-induced transitions from data, effectively overcoming limitations of existing methods. The multi-scaling reservoir computing approach, validated across diverse simulated and experimental systems, accurately predicts transition statistics and, crucially, even specific transition times in colored noise scenarios. Future research might explore enhancements through Bayesian optimization or simulated annealing for hyperparameter search, extending the method to systems with hidden nodes or other noise types, or incorporating conditional generative adversarial networks for noise modeling. The approach holds broad potential for analyzing complex noisy systems across diverse scientific disciplines.

The hyperparameter tuning process relies on a trial-and-error approach guided by convergence behavior and PSD matching, which could be improved by employing more sophisticated optimization techniques like Bayesian optimization. The method's performance might be affected by the specific characteristics of the noise distribution; additional testing on varied noise types is warranted. Although the method successfully captured protein folding dynamics, the generalizability to other types of biological systems or other complex systems with drastically different characteristics would benefit from further investigation.