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publications
Bayesian system ID: optimal management of parameter, model, and measurement uncertainty
Published in Nonlinear Dynamics, 2020
System identification of dynamical systems is often posed as a least squares minimization problem. The aim of these optimization problems is typically to learn either propagators or the underlying vector fields from trajectories of data. In this paper, we study a first principles derivation of appropriate objective formulations for system identification based on probabilistic principles. We compare the resulting inference objective to those used by emerging data-driven methods based on dynamic mode decomposition (DMD) and system identification of nonlinear dynamics (SINDy). We show that these and related least squares formulations are specific cases of a more general objective function. We also show that the more general objective function yields more robust and reliable recovery in the presence of sparse data and noisy measurements. We attribute this success to an explicit accounting of imperfect model forms, parameter uncertainty, and measurement uncertainty. We study the computational complexity of an approximate marginal Markov Chain Monte Carlo method to solve the resulting inference problem and numerically compare our results on a number of canonical systems: linear pendulum, nonlinear pendulum, the Van der Pol oscillator, the Lorenz system, and a reaction–diffusion system. The results of these comparisons show that in cases where DMD and SINDy excel, the Bayesian approach performs equally well, and in cases where DMD and SINDy fail to produce reasonable results, the Bayesian approach remains robust and can still deliver reliable results.
Recommended citation: Nicholas Galioto and Alex Arkady Gorodetsky. Bayesian system ID: optimal management of parameter, model, and measurement uncertainty. Nonlinear Dynamics, 102(1):241-267, 2020. https://link.springer.com/article/10.1007/s11071-020-05925-8
Bayesian identification of Hamiltonian dynamics from symplectic data
Published in 59th IEEE Conference on Decision and Control, 2020
We propose a Bayesian probabilistic formulation for system identification of Hamiltonian systems. This approach uses an approximate marginal Markov Chain Monte Carlo algorithm to directly discover a system Hamiltonian from data. Our approach improves upon existing methods in two ways: first we encode the fact that the data generating process is symplectic directly into our learning objective, and second we utilize a learning objective that simultaneously accounts for unknown parameters, model form, and measurement noise. This objective is the log marginal posterior of a probabilistic model that embeds a symplectic and reversible integrator within an uncertain dynamical system. We demonstrate that the resulting learning problem yields dynamical systems that have improved accuracy and reduced predictive uncertainty compared to existing state-of-the-art approaches. Simulation results are shown on the Hénon-Heiles Hamiltonian system.
Recommended citation: Nicholas Galioto and Alex Arkady Gorodetsky. Bayesian identification of Hamiltonian dynamics from symplectic data. In 2020 59th IEEE Conference on Decision and Control (CDC), pages 1190–1195. IEEE, 2020. https://ieeexplore.ieee.org/abstract/document/9303852
A New Objective for Identification of Partially Observed Linear Time-Invariant Dynamical Systems from Input-Output Data
Published in 3rd Conference on Learning for Dynamics and Control, 2021
In this work we consider the identification of partially observed dynamical systems from a single trajectory of arbitrary input-output data. We propose a new optimization objective, derived as a MAP estimator of a certain posterior, that explicitly accounts for model, measurement, and parameter uncertainty. This algorithm identifies a linear time invariant model on a hidden latent space of pre-specified dimension. In contrast to Markov-parameter based least squares approaches, our algorithm can be applied to systems with arbitrary forcing and initial conditions, and we empirically show several magnitude improvement in prediction quality compared to state-of-the-art approaches on both linear and nonlinear systems. Furthermore, we theoretically demonstrate how these existing approaches can be derived from simplifying assumptions on our system that neglect the possibility of model errors.
Recommended citation: Nicholas Galioto and Alex Arkady Gorodetsky. A New Objective for Identification of Partially Observed Linear Time-Invariant Dynamical Systems from Input-Output Data. In 3rd Conference on Learning for Dynamics and Control, pages 1180--1191. PMLR, 2021. https://proceedings.mlr.press/v144/galioto21a.html
Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models
Published in 61st IEEE Conference on Decision and Control, 2022
This paper proposes a probabilistic Bayesian formulation for system identification (ID) and estimation of nonseparable Hamiltonian systems using stochastic dynamic models. Nonseparable Hamiltonian systems arise in models from diverse science and engineering applications such as astrophysics, robotics, vortex dynamics, charged particle dynamics, and quantum mechanics. The numerical experiments demonstrate that the proposed method recovers dynamical systems with higher accuracy and reduced predictive uncertainty compared to state-of-the-art approaches. The results further show that accurate predictions far outside the training time interval in the presence of sparse and noisy measurements are possible, which lends robustness and generalizability to the proposed approach. A quantitative benefit is prediction accuracy with less than 10% relative error for more than 12 times longer than a comparable least-squares-based method on a benchmark problem.
Recommended citation: Harsh Sharma, Nicholas Galioto, Alex Arkady Gorodetsky, and Boris Kramer. Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models. In 2022 61st IEEE Conference on Decision and Control (CDC), pages 6742--6749. IEEE, 2022. https://ieeexplore.ieee.org/abstract/document/9992571
Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling
Published in arXiv, 2024
This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems using stochastic dynamic models allowing for statistically-dependent, vector-valued additive and multiplicative measurement noise. The approach is comprised of three main facets. First, we derive a Gaussian filter for a statistically-dependent, vector-valued, additive and multiplicative noise model that is needed to evaluate the likelihood within the Bayesian posterior. Second, we develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. Third, we demonstrate how structure-preserving methods can be incorporated into the proposed framework, using nonseparable Hamiltonians as an illustrative system class. We compare the Bayesian method to a state-of-the-art machine learning method on a canonical nonseparable Hamiltonian model and a chaotic double pendulum model with small, noisy training datasets. The results show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error using training data with up to 10% multiplicative noise compared to a standard training objective. Lastly, we demonstrate the utility of the novel algorithm for parameter estimation of a 64-dimensional model of the spatially-discretized nonlinear Schrödinger equation with data corrupted by up to 20% multiplicative noise.
Recommended citation: Nicholas Galioto, Harsh Sharma, Boris Kramer, and Alex Arkady Gorodetsky. Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling. arXiv preprint arXiv:2401.12476, 2024. https://arxiv.org/abs/2401.12476
Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification
Published in Physica D: Nonlinear Phenomena, 2024
This paper considers the problem of system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We propose and analyze an objective function derived from a Bayesian formulation for learning a hidden Markov model with stochastic dynamics. We then analyze this objective function in the context of several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that our proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics’ origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach can identify accurate and generalizable models even when there are more parameters than data or when the underlying system exhibits chaotic behavior.
Recommended citation: Nicholas Galioto and Alex Arkady Gorodetsky. Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification. Physica D: Nonlinear Phenomena, 462:134146, 2024. https://www.sciencedirect.com/science/article/pii/S0167278924000976
talks
Bayesian approaches for data-driven learning of dynamical systems
Published:
Poster presentation at 3rd Physics Informed Machine Learning Workshop. The poster can be found here.
Bayesian identification of Hamiltonian dynamics from symplectic data
Published:
Minisymposium presentation at 59th Conference on Decision and Control. The slides can be found here.
Robust Bayesian inference by accounting for model error: with applications to Hamiltonian systems models
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Minisymposium presentation at SIAM Conference on Computational Science and Engineering (CSE21). The slides can be found here.
A new objective function for identification of partially observed LTI dynamical systems from input-output data.
Published:
Poster presentation at 3rd Annual Learning for Dynamics & Control Conference. The poster can be found here.
Accounting for model errors in probabilistic linear identification of nonlinear PDE systems
Published:
Minisymposium presentation at 16th U.S. National Congress on Computational Mechanics. The slides can be found here.
Enforcing physical phenomena in system identification using Bayesian inference and stochastic models
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Conference presentation at IACM Conference on Mechanistic Machine Learning and Digital Twins for Computational Science, Engineering and Technology (MMLDT-CSET). The slides can be found here.
Bayesian learning of stochastic dynamical models for quantities of interest
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Minisymposium presentation at SIAM Conference on Uncertainty Quantification (UQ22). The slides can be found here.
Accounting for model uncertainty in the identification of partially known models
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Minisymposium presentation at 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). The slides can be found here.
Learning partially observed stochastic dynamical systems
Published:
Minisymposium presentation at SIAM Conference on Mathematics of Data Science (MDS22). The slides can be found here.
Correcting for Error in Reduced-Order Modeling Using Experimental Partial Observations and Bayesian System ID
Published:
Minisymposium presentation at SIAM Conference on Uncertainty Quantification (UQ24). The slides can be found here.
teaching
Teaching experience 1
Undergraduate course, University 1, Department, 2014
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Teaching experience 2
Workshop, University 1, Department, 2015
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