Generative learning for nonlinear dynamics

Modern generative machine learning models demonstrate surprising ability to create realistic outputs far beyond their training data, such as photorealistic artwork, accurate protein structures, or conversational text. These successes suggest that generative models learn to effectively parametrize and sample arbitrarily complex distributions. Beginning half a century ago, foundational works in nonlinear dynamics used tools from information theory to infer properties of chaotic attractors from time series, motivating the development of algorithms for parametrizing chaos in real datasets. In this perspective, we aim to connect these classical works to emerging themes in large-scale generative statistical learning. We first consider classical attractor reconstruction, which mirrors constraints on latent representations learned by state space models of time series. We next revisit early efforts to use symbolic approximations to compare minimal discrete generators underlying complex processes, a problem relevant to modern efforts to distill and interpret black-box statistical models. Emerging interdisciplinary works bridge nonlinear dynamics and learning theory, such as operator-theoretic methods for complex fluid flows, or detection of broken detailed balance in biological datasets. We anticipate that future machine learning techniques may revisit other classical concepts from nonlinear dynamics, such as transinformation decay and complexity-entropy tradeoffs.

Large statistical learning models effectively forecast diverse chaotic systems

Chaos and unpredictability are traditionally synonymous, yet recent advances in statistical forecasting suggest that large machine learning models can derive unexpected insight from extended observation of complex systems. Here, we study the forecasting of chaos at scale, by performing a large-scale comparison of 24 representative state-of-the-art multivariate forecasting methods on a crowdsourced database of 135 distinct low-dimensional chaotic systems. We find that large, domain-agnostic time series forecasting methods based on artificial neural networks consistently exhibit strong forecasting performance, in some cases producing accurate predictions lasting for dozens of Lyapunov times. Best-in-class results for forecasting chaos are achieved by recently-introduced hierarchical neural basis function models, though even generic transformers and recurrent neural networks perform strongly. However, physics-inspired hybrid methods like neural ordinary equations and reservoir computers contain inductive biases conferring greater data efficiency and lower training times in data-limited settings. We observe consistent correlation across all methods despite their widely-varying architectures, as well as universal structure in how predictions decay over long time intervals. Our results suggest that a key advantage of modern forecasting methods stems not from their architectural details, but rather from their capacity to learn the large-scale structure of chaotic attractors.

Recurrences reveal shared causal drivers of complex time series

Many experimental time series measurements share an unobserved causal driver. Examples include genes targeted by transcription factors, ocean flows influenced by large-scale atmospheric currents, and motor circuits steered by descending neurons. Reliably inferring this unseen driving force is necessary to understand the intermittent nature of top-down control schemes in diverse biological and engineered systems. Here, we introduce a new unsupervised learning algorithm that uses recurrences in time series measurements to gradually reconstruct an unobserved driving signal. Drawing on the mathematical theory of skew-product dynamical systems, we identify recurrence events shared across response time series, which implicitly define a recurrence graph with glass-like structure. As the amount or quality of observed data improves, this recurrence graph undergoes a percolation transition manifesting as weak ergodicity breaking for random walks on the induced landscape -- revealing the shared driver's dynamics, even in the presence of strongly corrupted or noisy measurements. Across several thousand random dynamical systems, we empirically quantify the dependence of reconstruction accuracy on the rate of information transfer from a chaotic driver to the response systems, and we find that effective reconstruction proceeds through gradual approximation of the driver's dominant unstable periodic orbits. Through extensive benchmarks against classical and neural-network-based signal processing techniques, we demonstrate our method's strong ability to extract causal driving signals from diverse real-world datasets spanning neuroscience, genomics, fluid dynamics, and physiology.

Chaos as an interpretable benchmark for forecasting and data-driven modelling

The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying attractor. Chaotic systems thus pose a unique challenge to modern statistical learning techniques, while retaining quantifiable mathematical properties that make them controllable and interpretable as benchmarks. Here, we present a growing database currently comprising 131 known chaotic dynamical systems spanning fields such as astrophysics, climatology, and biochemistry. Each system is paired with precomputed multivariate and univariate time series. Our dataset has comparable scale to existing static time series databases; however, our systems can be re-integrated to produce additional datasets of arbitrary length and granularity. Our dataset is annotated with known mathematical properties of each system, and we perform feature analysis to broadly categorize the diverse dynamics present across the collection. Chaotic systems inherently challenge forecasting models, and across extensive benchmarks we correlate forecasting performance with the degree of chaos present. We also exploit the unique generative properties of our dataset in several proof-of-concept experiments: surrogate transfer learning to improve time series classification, importance sampling to accelerate model training, and benchmarking symbolic regression algorithms.

Deep learning of dynamical attractors from time series measurements

Experimental measurements of physical systems often have a finite number of independent channels, causing essential dynamical variables to remain unobserved. However, many popular methods for unsupervised inference of latent dynamics from experimental data implicitly assume that the measurements have higher intrinsic dimensionality than the underlying system---making coordinate identification a dimensionality reduction problem. Here, we study the opposite limit, in which hidden governing coordinates must be inferred from only a low-dimensional time series of measurements. Inspired by classical techniques for studying the strange attractors of chaotic systems, we introduce a general embedding technique for time series, consisting of an autoencoder trained with a novel latent-space loss function. We first apply our technique to a variety of synthetic and real-world datasets with known strange attractors, and we use established and novel measures of attractor fidelity to show that our method successfully reconstructs attractors better than existing techniques. We then use our technique to discover dynamical attractors in datasets ranging from patient electrocardiograms, to household electricity usage, to eruptions of the Old Faithful geyser---demonstrating diverse applications of our technique for exploratory data analysis.

Cellular automata as convolutional neural networks

Deep learning techniques have recently demonstrated broad success in predicting complex dynamical systems ranging from turbulence to human speech, motivating broader questions about how neural networks encode and represent dynamical rules. We explore this problem in the context of cellular automata (CA), simple dynamical systems that are intrinsically discrete and thus difficult to analyze using standard tools from dynamical systems theory. We show that any CA may readily be represented using a convolutional neural network with a network-in-network architecture. This motivates our development of a general convolutional multilayer perceptron architecture, which we find can learn the dynamical rules for arbitrary CA when given videos of the CA as training data. In the limit of large network widths, we find that training dynamics are strongly stereotyped across replicates, and that common patterns emerge in the structure of networks trained on different CA rulesets. We train ensembles of networks on randomly-sampled CA, and we probe how the trained networks internally represent the CA rules using an information-theoretic technique based on distributions of layer activation patterns. We find that CA with simpler rule tables produce trained networks with hierarchical structure and layer specialization, while more complex CA tend to produce shallower representations---illustrating how the underlying complexity of the CA's rules influences the specificity of these internal representations. Our results suggest how the entropy of a physical process can affect its representation when learned by neural networks.