System Identification

Equation discovery is a fundamental learning task for uncovering the underlying dynamics of complex systems, with wide-ranging applications in areas such as brain connectivity analysis, climate modeling, gene regulation, and physical simulation. However, many existing approaches rely on accurate derivative estimation and are limited to first-order dynamical systems, restricting their applicability in real-world scenarios. In this work, we propose Sparse Equation Matching (SEM), a unified framework that encompasses several existing equation discovery methods under a common formulation. SEM introduces an integral-based sparse regression approach using Green's functions, enabling derivative-free estimation of differential operators and their associated driving functions in general-order dynamical systems. The effectiveness of SEM is demonstrated through extensive simulations, benchmarking its performance against derivative-based approaches. We then apply SEM to electroencephalographic (EEG) data recorded during multiple oculomotor tasks, collected from 52 participants in a brain-computer interface experiment. Our method identifies active brain regions across participants and reveals task-specific connectivity patterns. These findings offer valuable insights into brain connectivity and the underlying neural mechanisms.

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep learning models often require large datasets and significant computational resources for both training and inference due to their high-dimensional parameterizations. To address this challenge, we propose a meta-learning framework that discovers a low-dimensional manifold within the parameter space of an over-parameterized neural network architecture. This manifold is learned from an abundant meta-dataset of input-output sequences generated by a simulator of the class of systems of interest, and thereafter enables efficient model training while preserving the network's expressive power for the considered system class. Unlike bilevel meta-learning approaches, our method employs an auxiliary neural network to map datasets directly onto the learned manifold, eliminating the need for costly second-order gradient computations during meta-training and generally reducing the number of first-order updates required in inference, which could be expensive for large models. Finally, we derive a probabilistic bound that quantifies the performance gap between training on the learned low-dimensional manifold and the full parameterization. We validate our approach on a family of Bouc-Wen oscillators, which is a well-studied nonlinear system identification benchmark. We demonstrate that we are able to learn accurate models even in small-data scenarios.

Dynamical systems form the foundation of scientific discovery, traditionally modeled with predefined state variables such as the angle and angular velocity, and differential equations such as the equation of motion for a single pendulum. We introduce a framework that automatically discovers a low-dimensional and operable representation of system dynamics, including a set of compact state variables that preserve the smoothness of the system dynamics and a differentiable vector field, directly from video without requiring prior domain-specific knowledge. The prominence and effectiveness of the proposed approach are demonstrated through both quantitative and qualitative analyses of a range of dynamical systems, including the identification of stable equilibria, the prediction of natural frequencies, and the detection of chaotic and limit cycle behaviors. The results highlight the potential of our data-driven approach to advance automated scientific discovery.

This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to the well-understood learning theory in RKHSs under one unified framework. Consequently, our proposed framework allows for arbitrarily accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without a priori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.

Patterns of microcircuitry in the cerebral cortex suggest that the brain has an array of repeated canonical computational units. However, neural representations are distributed, so the relevant computations may only be related indirectly to single-neuron transformations. It thus remains an open challenge how to define canonical distributed computations. Here we integrate normative and algorithmic theories of neural computation to present a mathematical framework for inferring canonical distributed computations from large-scale neural activity patterns. At the normative level, we hypothesize that the brain learns a structured internal model of its environment and its latent causes. At the algorithmic level, we propose that this inference process is a nonlinear message-passing algorithm on a graph-structured model of the world. Given a time series of neural activity during a perceptual inference task, our analysis framework finds (i) the neural representation of the relevant latent variables, (ii) interactions between these latent variables that define the brain's internal model of the world, and (iii) message functions that specify the inference algorithm. With enough data, key computational properties are identifiable due to constraints from the symmetries inherent in any canonical computation, up to a joint global transformation of all interactions and the message-passing functions. As a demonstration of this framework, we analyze artificial neural recordings generated by a simulated brain that implicitly implements an approximate inference algorithm. Given simulated inputs and neural activity we successfully recover the latent variables, their neural representation and dynamics, and canonical message functions that govern the dynamics. Analysis of these models reveals features of experimental design required to extract canonical computations from neural data.

Is it possible to understand the intricacies of a dynamical system not solely from its input/output pattern, but also by observing the behavior of other systems within the same class? This central question drives the study presented in this paper. In response to this query, we introduce a novel paradigm for system identification, addressing two primary tasks: one-step-ahead prediction and multi-step simulation. Unlike conventional methods, we do not directly estimate a model for the specific system. Instead, we learn a meta model that represents a class of dynamical systems. This meta model is trained on a potentially infinite stream of synthetic data, generated by simulators whose settings are randomly extracted from a probability distribution. When provided with a context from a new system-specifically, an input/output sequence-the meta model implicitly discerns its dynamics, enabling predictions of its behavior. The proposed approach harnesses the power of Transformers, renowned for their in-context learning capabilities. For one-step prediction, a GPT-like decoder-only architecture is utilized, whereas the simulation problem employs an encoder-decoder structure. Initial experimental results affirmatively answer our foundational question, opening doors to fresh research avenues in system identification.