Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Dated: September 5, 2025) Finding the governing equations from data by sparse optimization has become a popular approach to deterministic modeling of dynamical systems. Considering the physical situations where the data can be imperfect due to disturbances and measurement errors, we show that for many chaotic systems, widely used sparse-optimization methods for discovering governing equations produce models that depend sensitively on the measurement procedure, yet all such models generate virtually identical chaotic attractors, leading to a striking limitation that challenges the conventional notion of equation-based modeling in complex dynamical systems. Calculating the Koopman spectra, we find that the different sets of equations agree in their large eigenvalues and the differences begin to appear when the eigenvalues are smaller than an equation-dependent threshold. The results suggest that finding the governing equations of the system and attempting to interpret them physically may lead to misleading conclusions. It would be more useful to work directly with the available data using, e.g., machine-learning methods. In physical science, a methodology of biblical importance is developing a quantitative model by extracting a set of governing equations from experimental data.

Accurate modeling is crucial in many engineering and scientific applications, yet obtaining a reliable process model for complex systems is often challenging. To address this challenge, we propose a novel framework, reservoir computing with unscented Kalman filtering (RCUKF), which integrates data-driven modeling via reservoir computing (RC) with Bayesian estimation through the unscented Kalman filter (UKF). The RC component learns the nonlinear system dynamics directly from data, serving as a surrogate process model in the UKF prediction step to generate state estimates in high-dimensional or chaotic regimes where nominal mathematical models may fail. Meanwhile, the UKF measurement update integrates real-time sensor data to correct potential drift in the data-driven model. We demonstrate RCUKF effectiveness on well-known benchmark problems and a real-time vehicle trajectory estimation task in a high-fidelity simulation environment.

The data-driven modeling of dynamical systems has become an essential tool for the construction of accurate computational models from real-world data. In this process, the inherent differential structures underlying the considered physical phenomena are often neglected making the reinterpretation of the learned models in a physically meaningful sense very challenging. In this work, we present three data-driven modeling approaches for the construction of dynamical systems with second-order differential structure directly from frequency domain data. Based on the second-order structured barycentric form, we extend the well-known Adaptive Antoulas-Anderson algorithm to the case of second-order systems. Depending on the available computational resources, we propose variations of the proposed method that prioritize either higher computation speed or greater modeling accuracy, and we present a theoretical analysis for the expected accuracy and performance of the proposed methods. Three numerical examples demonstrate the effectiveness of our new structured approaches in comparison to classical unstructured data-driven modeling.

Building Management System (BMS) through a data-driven method always faces data and model scalability issues. We propose a methodology to tackle the scalability challenges associated with the development of data-driven models for BMS by using Large Language Models (LLMs). LLMs' code generation adaptability can enable broader adoption of BMS by "automating the automation," particularly the data handling and data-driven modeling processes. In this paper, we use LLMs to generate code that processes structured data from BMS and build data-driven models for BMS's specific requirements. This eliminates the need for manual data and model development, reducing the time, effort, and cost associated with this process. Our hypothesis is that LLMs can incorporate domain knowledge about data science and BMS into data processing and modeling, ensuring that the data-driven modeling is automated for specific requirements of different building types and control objectives, which also improves accuracy and scalability. We generate a prompt template following the framework of Machine Learning Operations so that the prompts are designed to systematically generate Python code for data-driven modeling. Our case study indicates that bi-sequential prompting under the prompt template can achieve a high success rate of code generation and code accuracy, and significantly reduce human labor costs.

Abstract: Mathematical models are vital to the field of metrology, playing a key role in the derivation of measurement results and the calculation of uncertainties from measurement data, informed by an understanding of the measurement process. These models generally represent the correlation between the quantity being measured and all other pertinent quantities. Such relationships are used to construct measurement systems that can interpret measurement data to generate conclusions and predictions about the measurement system itself. Classic models are typically analytical, built on fundamental physical principles. However, the rise of digital technology, expansive sensor networks, and high-performance computing hardware have led to a growing shift towards data-driven methodologies. This trend is especially prominent when dealing with large, intricate networked sensor systems in situations where there is limited expert understanding of the frequently changing real-world contexts. Here, we demonstrate the variety of opportunities that data-driven modeling presents, and how they have been already implemented in various real-world applications.

MLMOD is a software package for incorporating machine learning approaches and models into simulations of microscale mechanics and molecular dynamics in LAMMPS. Recent machine learning approaches provide promising data-driven approaches for learning representations for system behaviors from experimental data and high fidelity simulations. The package faciliates learning and using data-driven models for (i) dynamics of the system at larger spatial-temporal scales (ii) interactions between system components, (iii) features yielding coarser degrees of freedom, and (iv) features for new quantities of interest characterizing system behaviors. MLMOD provides hooks in LAMMPS for (i) modeling dynamics and time-step integration, (ii) modeling interactions, and (iii) computing quantities of interest characterizing system states. The package allows for use of machine learning methods with general model classes including Neural Networks, Gaussian Process Regression, Kernel Models, and other approaches. Here we discuss our prototype C++/Python package, aims, and example usage. The package is integrated currently with the mesocale and molecular dynamics simulation package LAMMPS and PyTorch. For related papers, examples, updates, and additional information see https://github.com/atzberg/mlmod and http://atzberger.org/.

To model complex turbulent flow and heat transfer phenomena, this study aims to analyze and develop a reduced modeling approach based on artificial neural network (ANN) and wrapper methods. This approach has an advantage over other methods such as the correlation-based filter method in terms of removing redundant or irrelevant parameters even under non-linearity among them. As a downside, the overfitting and randomness of ANN training may produce inconsistent subsets over selection trials especially in a higher physical dimension. This study analyzes a few existing ANN-based wrapper methods and develops a revised one based on the gradient-based subset selection indices to minimize the loss in the total derivative or the directional consistency at each elimination step. To examine parameter reduction performance and consistency-over-trials, we apply these methods to a manufactured subset selection problem, modeling of the bubble size in a turbulent bubbly flow, and modeling of the spatially varying turbulent Prandtl number in a duct flow. It is found that the gradient-based subset selection to minimize the total derivative loss results in improved consistency-over-trials compared to the other ANN-based wrapper methods, while removing unnecessary parameters successfully. For the reduced turbulent Prandtl number model, the gradient-based subset selection improves the prediction in the validation case over the other methods. Also, the reduced parameter subsets show a slight increase in the training speed compared to the others.

The reliable prediction of the temporal behavior of complex systems is key in numerous scientific fields. This strong interest is however hindered by modeling issues: often, the governing equations describing the physics of the system under consideration are not accessible or, if known, their solution might require a computational time incompatible with the prediction time constraints. Not surprisingly, approximating complex systems in a generic functional format and informing it ex-nihilo from available observations has become common practice in the age of machine learning, as illustrated by the numerous successful examples based on deep neural networks. However, generalizability of the models, margins of guarantee and the impact of data are often overlooked or examined mainly by relying on prior knowledge of the physics. We tackle these issues from a different viewpoint, by adopting a curriculum learning strategy. In curriculum learning, the dataset is structured such that the training process starts from simple samples towards more complex ones in order to favor convergence and generalization. The concept has been developed and successfully applied in robotics and control of systems. Here, we apply this concept for the learning of complex dynamical systems in a systematic way. First, leveraging insights from the ergodic theory, we assess the amount of data sufficient for a-priori guaranteeing a faithful model of the physical system and thoroughly investigate the impact of the training set and its structure on the quality of long-term predictions. Based on that, we consider entropy as a metric of complexity of the dataset; we show how an informed design of the training set based on the analysis of the entropy significantly improves the resulting models in terms of generalizability, and provide insights on the amount and the choice of data required for an effective data-driven modeling.

Machine learning permeated almost every scientific discipline (Shanmuganathan, 2016; Wuest et al., 2016; Schmidt et al., 2019), and Solid Mechanics is no exception (Bessa et al., 2017; Capuano and Rimoli, 2019; Thakolkaran et al., 2022). With all their merits and flaws (Karniadakis et al., 2021), these algorithms provide a means to understand large datasets, finding patterns and modeling behavior where analytical solutions are challenging to obtain or not accurate enough. Focusing on plasticity modeling, its path-dependency posed a specific machine learning challenge that was recently addressed using recurrent neural networks, where time (or pseudo-time) can be naturally incorporated (Mozaffar et al., 2019). Since then several research groups proposed new neural network architectures and solved increasingly complex plasticity problems (Zhang and Mohr, 2020; Abueidda et al., 2021; Saidi et al., 2022; Bonatti et al., 2022). A similar trend is ongoing in other fields within and outside Mechanics (Liu et al., 2019a; Peng et al., 2021; Khalil et al., 2017; Dütting et al., 2019). Simultaneously, the scientific community is experiencing strong incentives to adhere to open science, with vehement support from funding agencies throughout the World to share data and models according to FAIR principles (Findable, Accessible, Interoperable and Reusable) (Wilkinson et al., 2016; Draxl and Scheffler, 2018; Jacobsen et al., 2020). There is also a clear need for end users to reuse these models and data. Nevertheless, there is a serious issue that obstructs the synergistic use of machine learning models by the community. Artificial neural networks, unlike biological neural networks, suffer from catastrophic forgetting (McCloskey and Cohen, 1989; French, 1999; Goodfellow et al., 2013).

Predicting the behaviors of pedestrian crowds is of critical importance for a variety of real-world problems. Data driven modeling, which aims to learn the mathematical models from observed data, is a promising tool to construct models that can make accurate predictions of such systems. In this work, we present a data-driven modeling approach based on the ODE-Net framework, for constructing continuous-time models of crowd dynamics. We discuss some challenging issues in applying the ODE-Net method to such problems, which are primarily associated with the dimensionality of the underlying crowd system, and we propose to address these issues by incorporating the social-force concept in the ODE-Net framework. Finally application examples are provided to demonstrate the performance of the proposed method.