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1. Data-driven model discovery and model selection for noisy biological systems

   https://doi.org/10.1371/journal.pcbi.1012762  

   Wu, Xiaojun; McDermott, MeiLu; MacLean, Adam L (January 2025, PLOS Computational Biology)

   Alber, Mark (Ed.)

   Biological systems exhibit complex dynamics that differential equations can often adeptly represent. Ordinary differential equation models are widespread; until recently their construction has required extensive prior knowledge of the system. Machine learning methods offer alternative means of model construction: differential equation models can be learnt from data via model discovery using sparse identification of nonlinear dynamics (SINDy). However, SINDy struggles with realistic levels of biological noise and is limited in its ability to incorporate prior knowledge of the system. We propose a data-driven framework for model discovery and model selection using hybrid dynamical systems: partial models containing missing terms. Neural networks are used to approximate the unknown dynamics of a system, enabling the denoising of the data while simultaneously learning the latent dynamics. Simulations from the fitted neural network are then used to infer models using sparse regression. We show, via model selection, that model discovery using hybrid dynamical systems outperforms alternative approaches. We find it possible to infer models correctly up to high levels of biological noise of different types. We demonstrate the potential to learn models from sparse, noisy data in application to a canonical cell state transition using data derived from single-cell transcriptomics. Overall, this approach provides a practical framework for model discovery in biology in cases where data are noisy and sparse, of particular utility when the underlying biological mechanisms are partially but incompletely known. 

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2. Physics-informed learning of governing equations from scarce data

   https://doi.org/10.1038/s41467-021-26434-1  

   Chen, Zhao; Liu, Yang; Sun, Hao (December 2021, Nature Communications)

   Abstract Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture. 

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3. TIME-SERIES FORECASTING AND REFINEMENT WITHIN A MULTIMODAL PDE FOUNDATION MODEL

   https://doi.org/10.1615/JMachLearnModelComput.2025057618  

   Jollie, Derek; Sun, Jingmin; Zhang, Zecheng; Schaeffer, Hayden (January 2025, Journal of Machine Learning for Modeling and Computing)

   Symbolic encoding has been used in multioperator learning (MOL) as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations (PDEs), the equation itself provides an additional modality to identify the system. The utilization of symbolic expressions alongside time-series samples allows for the development of multimodal predictive neural networks. A key challenge with current approaches is that the symbolic information, i.e., the equations, must be manually preprocessed (simplified, rearranged, etc.) to match and relate to the existing token library, which increases costs and reduces flexibility, especially when dealing with new differential equations. We propose a new token library based on SymPy to encode differential equations as an additional modality for time-series models. The proposed approach incurs minimal cost, is automated, and maintains high prediction accuracy for forecasting tasks. Additionally, we include a Bayesian filtering module that connects the different modalities to refine the learned equation. This improves the accuracy of the learned symbolic representation and the predicted time-series. 

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4. Symbolic regression via neural networks

   https://doi.org/10.1063/5.0134464  

   Boddupalli, N; Matchen, T; Moehlis, J (August 2023, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   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 and then show the accuracy of our algorithm across a range of classical dynamical systems. 

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5. A review on symbolic regression in power systems: Methods, applications, and future directions

   https://doi.org/10.1016/j.rser.2025.116075  

   Javadi, Amir Bahador; Pong, Philip (July 2025, Renewable and Sustainable Energy Reviews)

   As power systems evolve with the increasing integration of renewable energy sources and smart grid technologies, there is a growing demand for flexible and scalable modeling approaches capable of capturing the complex dynamics of modern grids. This review focuses on symbolic regression, a powerful methodology for deriving parsimonious and interpretable mathematical models directly from data. Symbolic regression is particularly valuable for power systems due to its ability to uncover governing equations without prior structural assumptions, enabling transparent and data-driven insights into nonlinear system behavior. The paper presents a comprehensive overview of symbolic regression methods, including sparse identification of nonlinear dynamics, automatic regression for governing equations, and deep symbolic regression, highlighting their applications in power systems. Through comparative case studies of the single machine infinite bus system, grid-following, and grid-forming inverters, we analyze the strengths, limitations, and suitability of each symbolic regression method in modeling nonlinear power system dynamics. Additionally, we identify critical research gaps and discuss future directions for leveraging symbolic regression in the optimization, control, and operation of modern power grids. This review aims to provide a valuable resource for researchers and engineers seeking innovative, data-driven solutions for modeling in the context of evolving power system infrastructure. 

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