Agent-based modeling (ABM) is a powerful tool for understanding self-organizing biological systems, but it is computationally intensive and often not analytically tractable. Equation learning (EQL) methods can derive continuum models from ABM data, but they typically require extensive simulations for each parameter set, raising concerns about generalizability. In this work, we extend EQL to Multi-experiment equation learning (ME-EQL) by introducing two methods: one-at-a-time ME-EQL (OAT ME-EQL), which learns individual models for each parameter set and connects them via interpolation, and embedded structure ME-EQL (ES ME-EQL), which builds a unified model library across parameters. We demonstrate these methods using a birth--death mean-field model and an on-lattice agent-based model of birth, death, and migration with spatial structure. Our results show that both methods significantly reduce the relative error in recovering parameters from agent-based simulations, with OAT ME-EQL offering better generalizability across parameter space. Our findings highlight the potential of equation learning from multiple experiments to enhance the generalizability and interpretability of learned models for complex biological systems.
Inverse problem or parameter estimation of ordinary differential equations is a process of obtaining the best parameters using experimental measurements of the states. Single (Multiple)-shooting is a type of sequential optimization method that minimizes the error in the measured and numerically integrated states. However, this requires computing sensitivities i.e. the derivatives of states with respect to the parameters over the numerical integrator, which can get computationally expensive. To address this challenge, many interpolation-based approaches have been proposed to either reduce the computational cost of sensitivity calculations or eliminate their need. In this paper, we use a bi-level optimization framework that leverages interpolation and exploits the structure of the differential equation to solve an inner convex optimization problem. We apply this method to two different problem formulations. First, parameter estimation for differential equations, and delayed differential equations, where the model structure is known but the parameters are unknown. Second, model discovery problems, where both the model structure and parameters are unknown.
Modeling propagation is the cornerstone for designing and optimizing next-generation wireless systems, with a particular emphasis on 5G and beyond era. Traditional modeling methods have long relied on statistic-based techniques to characterize propagation behavior across different environments. With the expansion of wireless communication systems, there is a growing demand for methods that guarantee the accuracy and interoperability of modeling. Artificial intelligence (AI)-based techniques, in particular, are increasingly being adopted to overcome this challenge, although the interpretability is not assured with most of these methods. Inspired by recent advancements in AI, this paper proposes a novel approach that accelerates the discovery of path loss models while maintaining interpretability. The proposed method automates the model formulation, evaluation, and refinement, facilitating model discovery. We evaluate two techniques: one based on Deep Symbolic Regression, offering full interpretability, and the second based on Kolmogorov-Arnold Networks, providing two levels of interpretability. Both approaches are evaluated on two synthetic and two real-world datasets. Our results show that Kolmogorov-Arnold Networks achieve R^2 values close to 1 with minimal prediction error, while Deep Symbolic Regression generates compact models with moderate accuracy. Moreover, on the selected examples, we demonstrate that automated methods outperform traditional methods, achieving up to 75% reduction in prediction errors, offering accurate and explainable solutions with potential to increase the efficiency of discovering next-generation path loss models.
Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by timescale separation, conservation laws, and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods to DAE systems with unknown constraints and time scales. We introduce Sparse Optimization for Differential-Algebraic Systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems and has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves numerical stability when handling high correlations between library terms -- caused by near-perfect algebraic relationships -- by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical, and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.
Theory of Mind (ToM), the ability to understand people's mental variables based on their behavior, is key to developing socially intelligent agents. Current approaches to Theory of Mind reasoning either rely on prompting Large Language Models (LLMs), which are prone to systematic errors, or use rigid, handcrafted Bayesian Theory of Mind (BToM) models, which are more robust but cannot generalize across different domains. In this work, we introduce AutoToM, an automated Bayesian Theory of Mind method for achieving open-ended machine Theory of Mind. AutoToM can operate in any domain, infer any mental variable, and conduct robust Theory of Mind reasoning of any order. Given a Theory of Mind inference problem, AutoToM first proposes an initial BToM model. It then conducts automated Bayesian inverse planning based on the proposed model, leveraging an LLM as the backend. Based on the uncertainty of the inference, it iteratively refines the model, by introducing additional mental variables and/or incorporating more timesteps in the context. Empirical evaluations across multiple Theory of Mind benchmarks demonstrate that AutoToM consistently achieves state-of-the-art performance, offering a scalable, robust, and interpretable approach to machine Theory of Mind.
Symbolic regression (SR) is an emerging branch of machine learning focused on discovering simple and interpretable mathematical expressions from data. Although a wide-variety of SR methods have been developed, they often face challenges such as high computational cost, poor scalability with respect to the number of input dimensions, fragility to noise, and an inability to balance accuracy and complexity. This work introduces SyMANTIC, a novel SR algorithm that addresses these challenges. SyMANTIC efficiently identifies (potentially several) low-dimensional descriptors from a large set of candidates (from $\sim 10^5$ to $\sim 10^{10}$ or more) through a unique combination of mutual information-based feature selection, adaptive feature expansion, and recursively applied $\ell_0$-based sparse regression. In addition, it employs an information-theoretic measure to produce an approximate set of Pareto-optimal equations, each offering the best-found accuracy for a given complexity. Furthermore, our open-source implementation of SyMANTIC, built on the PyTorch ecosystem, facilitates easy installation and GPU acceleration. We demonstrate the effectiveness of SyMANTIC across a range of problems, including synthetic examples, scientific benchmarks, real-world material property predictions, and chaotic dynamical system identification from small datasets. Extensive comparisons show that SyMANTIC uncovers similar or more accurate models at a fraction of the cost of existing SR methods.
Understanding the world and explaining it with scientific theories is a central aspiration of artificial intelligence research. Proposing theories, designing experiments to test them, and then revising them based on data are fundamental to scientific discovery. Despite the significant promise of LLM-based scientific agents, no benchmarks systematically test LLM's ability to propose scientific models, collect experimental data, and revise them in light of new data. We introduce BoxingGym, a benchmark with 10 environments for systematically evaluating both experimental design (e.g. collecting data to test a scientific theory) and model discovery (e.g. proposing and revising scientific theories). To enable tractable and quantitative evaluation, we implement each environment as a generative probabilistic model with which a scientific agent can run interactive experiments. These probabilistic models are drawn from various real-world scientific domains ranging from psychology to ecology. To quantitatively evaluate a scientific agent's ability to collect informative experimental data, we compute the expected information gain (EIG), an information-theoretic quantity which measures how much an experiment reduces uncertainty about the parameters of a generative model. A good scientific theory is a concise and predictive explanation. Therefore, to quantitatively evaluate model discovery, we ask a scientific agent to explain their model and then assess whether this explanation enables another scientific agent to make reliable predictions about this environment. In addition to this explanation-based evaluation, we compute standard model evaluation metrics such as prediction errors. We find that current LLMs, such as GPT-4o, struggle with both experimental design and model discovery. We find that augmenting the LLM-based agent with an explicit statistical model does not reliably improve these results.
Symbolic regression (SR) is a powerful machine learning approach that searches for both the structure and parameters of algebraic models, offering interpretable and compact representations of complex data. Unlike traditional regression methods, SR explores progressively complex feature spaces, which can uncover simple models that generalize well, even from small datasets. Among SR algorithms, the Sure Independence Screening and Sparsifying Operator (SISSO) has proven particularly effective in the natural sciences, helping to rediscover fundamental physical laws as well as discover new interpretable equations for materials property modeling. However, its widespread adoption has been limited by performance inefficiencies and the challenges posed by its FORTRAN-based implementation, especially in modern computing environments. In this work, we introduce TorchSISSO, a native Python implementation built in the PyTorch framework. TorchSISSO leverages GPU acceleration, easy integration, and extensibility, offering a significant speed-up and improved accuracy over the original. We demonstrate that TorchSISSO matches or exceeds the performance of the original SISSO across a range of tasks, while dramatically reducing computational time and improving accessibility for broader scientific applications.
We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.
Hyperspectral Imaging (HSI) plays an increasingly critical role in precise vision tasks within remote sensing, capturing a wide spectrum of visual data. Transformer architectures have significantly enhanced HSI task performance, while advancements in Transformer Architecture Search (TAS) have improved model discovery. To harness these advancements for HSI classification, we make the following contributions: i) We propose HyTAS, the first benchmark on transformer architecture search for Hyperspectral imaging, ii) We comprehensively evaluate 12 different methods to identify the optimal transformer over 5 different datasets, iii) We perform an extensive factor analysis on the Hyperspectral transformer search performance, greatly motivating future research in this direction. All benchmark materials are available at HyTAS.
There exist endless examples of dynamical systems with vast available data and unsatisfying mathematical descriptions. Sparse regression applied to symbolic libraries has quickly emerged as a powerful tool for learning governing equations directly from data; these learned equations balance quantitative accuracy with qualitative simplicity and human interpretability. Here, I present a general purpose, model agnostic sparse regression algorithm that extends a recently proposed exhaustive search leveraging iterative Singular Value Decompositions (SVD). This accelerated scheme, Scalable Pruning for Rapid Identification of Null vecTors (SPRINT), uses bisection with analytic bounds to quickly identify optimal rank-1 modifications to null vectors. It is intended to maintain sensitivity to small coefficients and be of reasonable computational cost for large symbolic libraries. A calculation that would take the age of the universe with an exhaustive search but can be achieved in a day with SPRINT.
We propose a new parameter-adaptive uncertainty-penalized Bayesian information criterion (UBIC) to prioritize the parsimonious partial differential equation (PDE) that sufficiently governs noisy spatial-temporal observed data with few reliable terms. Since the naive use of the BIC for model selection has been known to yield an undesirable overfitted PDE, the UBIC penalizes the found PDE not only by its complexity but also the quantified uncertainty, derived from the model supports' coefficient of variation in a probabilistic view. We also introduce physics-informed neural network learning as a simulation-based approach to further validate the selected PDE flexibly against the other discovered PDE. Numerical results affirm the successful application of the UBIC in identifying the true governing PDE. Additionally, we reveal an interesting effect of denoising the observed data on improving the trade-off between the BIC score and model complexity. Code is available at https://github.com/Pongpisit-Thanasutives/UBIC.
While tumor dynamic modeling has been widely applied to support the development of oncology drugs, there remains a need to increase predictivity, enable personalized therapy, and improve decision-making. We propose the use of Tumor Dynamic Neural-ODE (TDNODE) as a pharmacology-informed neural network to enable model discovery from longitudinal tumor size data. We show that TDNODE overcomes a key limitation of existing models in its ability to make unbiased predictions from truncated data. The encoder-decoder architecture is designed to express an underlying dynamical law which possesses the fundamental property of generalized homogeneity with respect to time. Thus, the modeling formalism enables the encoder output to be interpreted as kinetic rate metrics, with inverse time as the physical unit. We show that the generated metrics can be used to predict patients' overall survival (OS) with high accuracy. The proposed modeling formalism provides a principled way to integrate multimodal dynamical datasets in oncology disease modeling.
This study presents a systematic machine learning approach for creating efficient hybrid models and discovering sorption uptake models in non-linear advection-diffusion-sorption systems. It demonstrates an effective method to train these complex systems using gradient based optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, combined with spatial discretization and adaptive integrators. Sparse and symbolic regression were employed to identify missing functions in the artificial neural network. The robustness of the proposed method was tested on an in-silico data set of noisy breakthrough curve observations of fixed-bed adsorption, resulting in a well-fitted hybrid model. The study successfully reconstructed sorption uptake kinetics using sparse and symbolic regression, and accurately predicted breakthrough curves using identified polynomials, highlighting the potential of the proposed framework for discovering sorption kinetic law structures.
We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data is generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.
For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behavior of natural and man-made materials in response to mechanical loading. Now, computer science offers a universal solution: Neural Networks. Neural Networks are powerful function approximators that can learn constitutive relations from large data without any knowledge of the underlying physics. However, classical Neural Networks ignore a century of research in constitutive modeling, violate thermodynamic considerations, and fail to predict the behavior outside the training regime. Here we design a new family of Constitutive Artificial Neural Networks that inherently satisfy common kinematic, thermodynamic, and physic constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. We revisit the non-linear field theories of mechanics and reverse-engineer the network input to account for material objectivity, symmetry, and incompressibility; the network output to enforce thermodynamic consistency; the activation functions to implement physically reasonable restrictions; and the network architecture to ensure polyconvexity. We demonstrate that this new class of models is a generalization of the classical neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a clear physical interpretation. When trained with classical benchmark data for rubber, our network autonomously selects the best constitutive model and learns its parameters. Our findings suggests that Constitutive Artificial Neural Networks have the potential to induce a paradigm shift in constitutive modeling, from user-defined model selection to automated model discovery. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.
Discovering governing equations of complex dynamical systems directly from data is a central problem in scientific machine learning. In recent years, the sparse identification of nonlinear dynamics (SINDy) framework, powered by heuristic sparse regression methods, has become a dominant tool for learning parsimonious models. We propose an exact formulation of the SINDy problem using mixed-integer optimization (MIO) to solve the sparsity constrained regression problem to provable optimality in seconds. On a large number of canonical ordinary and partial differential equations, we illustrate the dramatic improvement of our approach in accurate model discovery while being more sample efficient, robust to noise, and flexible in accommodating physical constraints.
Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. The choice of method depends on one's intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., modeling approaches using the suite of data-driven modeling methods on three continuous dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.
Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.
We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system x' = f(x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate x'. These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g. 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1% - 3% (for 50% noise), 6% - 8% (for 100% noise); and 23% - 25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on x' = f(x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.