Model predictive control (MPC), widely used for real-time control of complex dynamical systems, operates by repeatedly solving an optimization problem over a receding time horizon. Its success hinges on dynamical models that are accurate yet efficient enough for rapid online computation. Frequently, the governing models of complex systems are either unknown or computationally inefficient, forcing MPC to rely on data-driven surrogate models. Echo state networks (ESNs), a class of recurrent neural networks trained through computationally efficient ridge regression, are well-suited for this role and have demonstrated strong forecasting capabilities in chaotic dynamical systems. Their architecture naturally supports rapid training and flexible adaptation to varying control inputs. In this work, we demonstrate that ESNs serve as effective data-driven surrogates for system dynamics under diverse control scenarios, outperforming competing architectures such as long short-term memory (LSTM) networks. On challenging control benchmarks, including the Lorenz system with control and fluid flow past a cylinder, MPC with ESN surrogates consistently achieves the control objective, whereas the next-best considered architecture, LSTM-based MPC, frequently fails. Even in cases where LSTM-based MPC succeeds, ESN-based MPC reduces average control cost by up to 10% and decreases variability by as much as 85%. Beyond performance, ESNs are significantly more sample-efficient and train over an order of magnitude faster than LSTMs. These results establish ESNs as accurate, efficient architectures for scalable data-driven MPC in complex systems with limited training data and unknown dynamics.

Machine learning (ML) is transforming modeling and control in the physical, engineering, and biological sciences. However, rapid development has outpaced the creation of standardized, objective benchmarks—leading to weak baselines, reporting bias, and inconsistent evaluations across methods. This undermines reproducibility, misguides resource allocation, and obscures scientific progress. To address this, we propose a Common Task Framework (CTF) for scientific machine learning. The CTF features a curated set of datasets and task-specific metrics spanning forecasting, state reconstruction, and generalization under realistic constraints, including noise and limited data. Inspired by the success of CTFs in fields like natural language processing and computer vision, our framework provides a structured, rigorous foundation for head-to-head evaluation of diverse algorithms. As a first step, we benchmark methods on two canonical nonlinear systems: Kuramoto-Sivashinsky and Lorenz. These results illustrate the utility of the CTF in revealing method strengths, limitations, and suitability for specific classes of problems and diverse objectives. Next, we are launching a competition around a global real world sea surface temperature dataset with a true holdout dataset to foster community engagement. Our long-term vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets that raise the bar for rigor and reproducibility in scientific ML.

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.

Insect flight is a strongly nonlinear and actuated dynamical system. As such, strategies for understanding its control have typically relied on either model-based methods or linearizations thereof. Here we develop a framework that combines model predictive control on an established flight dynamics model and deep neural networks (DNN) to create an efficient method for solving the inverse problem of flight control. We turn to natural systems for inspiration since they inherently demonstrate network pruning with the consequence of yielding more efficient networks for a specific set of tasks. This bio-inspired approach allows us to leverage network pruning to optimally sparsify a DNN architecture in order to perform flight tasks with as few neural connections as possible, however, there are limits to sparsification. Specifically, as the number of connections falls below a critical threshold, flight performance drops considerably. We develop sparsification paradigms and explore their limits for control tasks. Monte Carlo simulations also quantify the statistical distribution of network weights during pruning given initial random weights of the DNNs. We demonstrate that on average, the network can be pruned to retain a small amount of original network weights and still perform comparably to its fully-connected counterpart. The relative number of remaining weights, however, is highly dependent on the initial architecture and size of the network. Overall, this work shows that sparsely connected DNNs are capable of predicting the forces required to follow flight trajectories. Additionally, sparsification has sharp performance limits.

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.

We propose a new low-rank tensor factorization where one mode is coded as a sparse linear combination of elements from an over-complete library. Our method, Shape Constrained Tensor Decomposition (SCTD) is based upon the CANDECOMP/PARAFAC (CP) decomposition which produces r-rank approximations of data tensors via outer products of vectors in each dimension of the data. The SCTD model can leverage prior knowledge about the shape of factors along a given mode, for example in tensor data where one mode represents time. By constraining the vector in the temporal dimension to known analytic forms which are selected from a large set of candidate functions, more readily interpretable decompositions are achieved and analytic time dependencies discovered. The SCTD method circumvents traditional flattening techniques where an N-way array is reshaped into a matrix in order to perform a singular value decomposition. A clear advantage of the SCTD algorithm is its ability to extract transient and intermittent phenomena which is often difficult for SVD-based methods. We motivate the SCTD method using several intuitively appealing results before applying it on a real-world data set to illustrate the efficiency of the algorithm in extracting interpretable spatio-temporal modes. With the rise of data-driven discovery methods, the decomposition proposed provides a viable technique for analyzing multitudes of data in a more comprehensible fashion.

Air quality models are limited by the computational costs associated with the simulation of the complex chemical and dynamical processes of reactive pollutants in the atmosphere. We discuss here the potential usage of machine learning and reduced-order modeling techniques to mitigate some of these limitations. We first give an overview of three new methods emerging from the field of signal processing - sparse sampling, randomized matrix decompositions and the construction of reduced order models - and discuss them in the context of air quality modeling. In the second part we discuss the substitution of the standard chemical solver of the chemistry model with a random forest regression model trained through machine learning. We find that this approach shows promising initial results for important air pollutants such as ozone (O3), predicting concentrations that deviate less than 10% from the values computed by the traditional model. The here highlighted methods all have the potential to significantly reduce the computational burden of air quality models while maintaining the model's capability to capture all features relevant to air quality. Such lightweight air quality models offer new opportunities for air quality forecasting and to assimilate the rapidly increasing array of air quality observations.

Using a model for the dynamics of the full somatic nervous system of the nematode C. elegans, we address how biological network architectures and their functionality are degraded in the presence of focal axonal swellings (FAS) arising from neurodegenerative disease and/or traumatic brain injury. Using biophysically measured FAS distributions and swelling sizes, we are able to simulate the effects of injuries on the neural dynamics of C. elegans, showing how damaging the network degrades its low-dimensional dynamical responses. We visualize these injured neural dynamics by mapping them onto the worm's low-dimensional postures, i.e. eigenworm modes. We show that a diversity of functional deficits arise from the same level of injury on a connectomic network. Functional deficits are quantified using a statistical shape analysis, a procrustes analysis, for deformations of the limit cycles that characterize key behaviors such as forward crawling. This procrustes metric carries information on the functional outcome of injuries in the model. Furthermore, we apply classification trees to relate injury structure to the behavioral outcome. This makes testable predictions for the structure of an injury given a defined functional deficit. More critically, this study demonstrates the potential role of computational simulation studies in understanding how neuronal networks process biological signals, and how this processing is impacted by network injury.

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.

Herpes simplex virus-2 (HSV-2) is a chronic reactivating infection that leads to recurrent shedding episodes in the genital tract. A minority of episodes are prolonged, and associated with development of painful ulcers. However, currently, available tools poorly predict viral trajectories and timing of reactivations in infected individuals. We employed principal components analysis (PCA) and singular value decomposition (SVD) to interpret HSV-2 genital tract shedding time series data, as well as simulation output from a stochastic spatial mathematical model. Empirical and model-derived, time-series data gathered over >30 days consists of multiple complex episodes that could not be reduced to a manageable number of descriptive features with PCA and SVD. However, single HSV-2 shedding episodes, even those with prolonged duration and complex morphologies consisting of multiple erratic peaks, were consistently described using a maximum of four dominant features. Modeled and clinical episodes had equivalent distributions of dominant features, implying similar dynamics in real and simulated episodes. We applied linear discriminant analysis (LDA) to simulation output and identified that local immune cell density at the viral reactivation site had a predictive effect on episode duration, though longer term shedding suggested chaotic dynamics and could not be predicted based on spatial patterns of immune cell density. These findings suggest that HSV-2 shedding patterns within an individual are impossible to predict over weeks or months, and that even highly complex single HSV-2 episodes can only be partially predicted based on spatial distribution of immune cell density.

Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability.