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Jeff Moehlis

Jeff Moehlis appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.DSMachine Learningcond-mat.othereess.SPmath.OCnlin.AO

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Published work

7 published item(s)

preprint2026arXiv

Symbolic Regression via Neural Networks

Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their capabilities in approximating dynamics from data, but a shortcoming of traditional deep learning is that there is little insight into the underlying mapping beyond its numerical output for a given input. This limits their utility in analysis beyond simple prediction. Simultaneously, a number of strategies exist which identify models based on a fixed dictionary of basis functions, but most either require some intuition or insight about the system, or are susceptible to overfitting or a lack of parsimony. Here we present a novel approach that combines the flexibility and accuracy of deep learning approaches with the utility of symbolic solutions: a deep neural network that generates a symbolic expression for the governing equations. We first describe the architecture for our model, then show the accuracy of our algorithm across a range of classical dynamical systems.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Analysis of Neural Clusters due to Deep Brain Stimulation Pulses

Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson's disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, theoretical studies of populations of neural oscillators stimulated by periodic pulses suggest that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters, their stability properties, and their basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Augmented Phase Reduction for Periodic Orbits Near a Homoclinic Bifurcation and for Relaxation Oscillators

Oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. The standard phase reduction, a model reduction technique based on isochrons, can be unsuitable for oscillators which have a small-magnitude negative nontrivial Floquet exponent. This necessitates the use of the augmented phase reduction, a recently devised two-dimensional reduction technique based on isochrons and isostables. In this article, we calculate analytical expressions for the augmented phase reduction for two dynamically different planar systems: periodic orbits born out of a homoclinic bifurcation, and relaxation oscillators. To validate our calculations, we simulate models in these dynamic regimes, and compare their numerically computed augmented phase reduction with the derived analytical expressions. These analytical and numerical calculations help us to understand conditions for which the use of augmented phase reduction over the standard phase reduction can be advantageous.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Supervised Learning Algorithms for Controlling Underactuated Dynamical Systems

Control of underactuated dynamical systems has been studied for decades in robotics, and is now emerging in other fields such as neuroscience. Most of the advances have been in model based control theory, which has limitations when the system under study is very complex and it is not possible to construct a model. This calls for data driven control methods like machine learning, which has spread to many fields in the recent years including control theory. However, the success of such algorithms has been dependent on availability of large datasets. Moreover, due to their black box nature, it is challenging to analyze how such algorithms work, which may be crucial in applications where failure is very costly. In this paper, we develop two related novel supervised learning algorithms. The algorithms are powerful enough to control a wide variety of complex underactuated dynamical systems, and yet have a simple and intelligent structure that allows them to work with a sparse data set even in the presence of noise. Our algorithms output a bang-bang (binary) control input by taking in feedback of the state of the dynamical system. The algorithms learn this control input by maximizing a reward function in both short and long time horizons. We demonstrate the versatility of our algorithms by applying them to a diverse range of applications including: switching between bistable states, changing the phase of an oscillator, desynchronizing a population of synchronized coupled oscillators, and stabilizing an unstable fixed point. For most of these applications we are able to reason why our algorithms work by using traditional dynamical systems and control theory. We also compare our learning algorithms with some traditional control algorithms, and reason why our algorithms work better.

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preprint2014arXiv

A Nonparametric Adaptive Nonlinear Statistical Filter

We use statistical learning methods to construct an adaptive state estimator for nonlinear stochastic systems. Optimal state estimation, in the form of a Kalman filter, requires knowledge of the system's process and measurement uncertainty. We propose that these uncertainties can be estimated from (conditioned on) past observed data, and without making any assumptions of the system's prior distribution. The system's prior distribution at each time step is constructed from an ensemble of least-squares estimates on sub-sampled sets of the data via jackknife sampling. As new data is acquired, the state estimates, process uncertainty, and measurement uncertainty are updated accordingly, as described in this manuscript.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics

For asymptotically periodic systems, a powerful (phase) reduction of the dynamics is obtained by computing the so-called isochrons, i.e. the sets of points that converge toward the same trajectory on the limit cycle. Motivated by the analysis of excitable systems, a similar reduction has been attempted for non-periodic systems admitting a stable fixed point. In this case, the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories. Instead, the sets of interest-that we call isostables-are defined in literature as the sets of points that converge toward the same trajectory on the stable slow manifold of the fixed point. However, it turns out that this definition of the isostables holds only for systems with slow-fast dynamics. Also, efficient methods for computing the isostables are missing. The present paper provides a general framework for the definition and the computation of the isostables of stable fixed points, which is based on the spectral properties of the so-called Koopman operator. More precisely, the isostables are defined as the level sets of a particular eigenfunction of the Koopman operator. Through this approach, the isostables are unique and well-defined objects related to the asymptotic properties of the system. Also, the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action-angle coordinates for the dynamics. In addition, an efficient algorithm for computing the isostables is obtained, which relies on the evaluation of Laplace averages along the trajectories. The method is illustrated with the excitable FitzHugh-Nagumo model and with the Lorenz model. Finally, we discuss how these methods based on the Koopman operator framework relate to the global linearization of the system and to the derivation of special Lyapunov functions.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Phase noise of oscillators with unsaturated amplifiers

We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operational point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback.

0 citations0 reviewsNo code linkedOpen access