Symmetry is present throughout nature and continues to play an increasingly central role in machine learning. In this paper, we provide a unifying theoretical and methodological framework for incorporating Lie group symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and 3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified candidate group only when the data provide sufficient evidence. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual under the bilinear pairing induced by the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularizers, built from the Lie derivative with a nuclear-norm relaxation, that penalizes symmetry breaking during training. We explain how these ideas can be applied to a wide range of machine learning models including basis-function regression, dynamical-systems discovery, neural networks, and neural operators acting on fields.

Single-pixel imaging is an efficient image acquisition process where light from a target scene is passed through a spatial light modulator and then projected onto a single photodiode with a high temporal acquisition rate. The scene reconstruction is achieved using computational methods that leverage prior assumptions on the scene structure. In this paper, we propose to model the structure of a dynamic spatio-temporal scene using a reduced-order model that is learned from training data examples. Specifically, by combining single-pixel imaging methods with a reduced-order model prior implemented as a neural ordinary differential equation, image sequence reconstruction can be accomplished with significantly reduced data requirements while maintaining performance levels on par with leading methods. We demonstrate superior reconstruction at low sampling rates for simulated trajectories governed by Burgers’ equation and turbulent plumes emulating gas leaks.

The dynamic mode decomposition (DMD) is a powerful data-driven modeling technique that reveals coherent spatiotemporal patterns from dynamical system snapshot observations. PyDMD is a Python package that implements DMD and several of its major optimizations and methodological extensions. In this paper, we introduce the version 1.0 release of PyDMD, which includes new data preprocessors, plotting tools, and a number of cutting-edge DMD methods specifically designed to handle real-world data that may be noisy, multi-scale, parameterized, prohibitively high-dimensional, and even strongly nonlinear. The package is friendly to install, thoroughly-documented, supplemented with extensive code examples, and modularly-structured to support future additions. The entire codebase is released under the MIT license and is available at https://github.com/PyDMD/PyDMD.

Accurate and agile trajectory tracking in sub-gram Micro Aerial Vehicles (MAVs) is challenging, as the small scale of the robot induces large model uncertainties, demanding robust feedback controllers, while the fast dynamics and computational constraints prevent the deployment of computationally expensive strategies. In this work, we present an approach for agile and computationally efficient trajectory tracking on the MIT SoftFly [1], a sub-gram MAV (0.7 grams). Our strategy employs a cascaded control scheme, where an adaptive attitude controller is combined with a neural network (NN) policy trained to imitate a trajectory tracking robust tube model predictive controller (RTMPC). The NN policy is obtained using our recent work [2], which enables the policy to preserve the robustness of RTMPC, but at a fraction of its computational cost. We experimentally evaluate our approach, achieving position Root Mean Square Errors (RMSEs) lower than 1.8 cm even in the more challenging maneuvers, obtaining a 60% reduction in maximum position error compared to [3], and demonstrating robustness to large external disturbances.

High-dimensional spatio-temporal dynamics can often be encoded in a low-dimensional subspace. Engineering applications for modeling, characterization, design, and control of such large-scale systems often rely on dimensionality reduction to make solutions computationally tractable in real time. Common existing paradigms for dimensionality reduction include linear methods, such as the singular value decomposition (SVD), and nonlinear methods, such as variants of convolutional autoencoders (CAE). However, these encoding techniques lack the ability to efficiently represent the complexity associated with spatio-temporal data, which often requires variable geometry, non-uniform grid resolution, adaptive meshing, and/or parametric dependencies. To resolve these practical engineering challenges, we propose a general framework called Neural Implicit Flow (NIF) that enables a mesh-agnostic, low-rank representation of large-scale, parametric, spatial-temporal data. NIF consists of two modified multilayer perceptrons (MLPs): (i) ShapeNet, which isolates and represents the spatial complexity, and (ii) ParameterNet, which accounts for any other input complexity, including parametric dependencies, time, and sensor measurements. We demonstrate the utility of NIF for parametric surrogate modeling, enabling the interpretable representation and compression of complex spatio-temporal dynamics, efficient many-spatial-query tasks, and improved generalization performance for sparse reconstruction.

We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods. We extensively benchmark these algorithms against other leading forecasting methods on a range of synthetic experiments as well as in the context of real-world power systems and fluid flows.

Retinal template matching and registration is an important challenge in teleophthalmology with low-cost imaging devices. However, the images from such devices generally have a small field of view (FOV) and image quality degradations, making matching difficult. In this paper, we develop an efficient and accurate retinal matching technique that combines dimension reduction and mutual information (MI), called RetinaMatch. The dimension reduction initializes the MI optimization as a coarse localization process, which narrows the optimization domain and avoids local optima. The effectiveness of RetinaMatch is demonstrated on the open fundus image database STARE with simulated reduced FOV and anticipated degradations, and on retinal images acquired by adapter-based optics attached to a smartphone. RetinaMatch achieves a success rate over 94% on human retinal images with the matched target registration errors below 2 pixels on average, excluding the observer variability, outperforming standard template matching solutions. In the application of measuring vessel diameter repeatedly, single pixel errors are expected. In addition, our method can be used in the process of image mosaicking with area-based registration, providing a robust approach when feature-based methods fail. To the best of our knowledge, this is the first template matching algorithm for retina images with small template images from unconstrained retinal areas. In the context of the emerging mixed reality market, we envision automated retinal image matching and registration methods as transformative for advanced teleophthalmology and long-term retinal monitoring.

Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent topological features that can be used for any supervised or unsupervised learning task, including multi-way classification. Sparse sampling, on the other hand, provides a highly efficient technique to reconstruct signals in the spatial-temporal domain from just a few carefully-chosen samples. Here, we present a new method, referred to as the Sparse-TDA algorithm, that combines favorable aspects of the two techniques. This combination is realized by selecting an optimal set of sparse pixel samples from the persistent features generated by a vector-based TDA algorithm. These sparse samples are selected from a low-rank matrix representation of persistent features using QR pivoting. We show that the Sparse-TDA method demonstrates promising performance on three benchmark problems related to human posture recognition and image texture classification.

Using a computational model of the Caenorhabditis elegans connectome dynamics, we show that proprioceptive feedback is necessary for sustained dynamic responses to external input. This is consistent with the lack of biophysical evidence for a central pattern generator, and recent experimental evidence that proprioception drives locomotion. The low-dimensional functional response of the Caenorhabditis elegans network of neurons to proprioception-like feedback is optimized by input of specific spatial wavelengths which correspond to the spatial scale of real body shape dynamics. Furthermore, we find that the motor subcircuit of the network is responsible for regulating this response, in agreement with experimental expectations. To explore how the connectomic dynamics produces the observed two-mode, oscillatory limit cycle behavior from a static fixed point, we probe the fixed point's low-dimensional structure using Dynamic Mode Decomposition. This reveals that the nonlinear network dynamics encode six clusters of dynamic modes, with timescales spanning three orders of magnitude. Two of these six dynamic mode clusters correspond to previously-discovered behavioral modes related to locomotion. These dynamic modes and their timescales are encoded by the network's degree distribution and specific connectivity. This suggests that behavioral dynamics are partially encoded within the connectome itself, the connectivity of which facilitates proprioceptive control.