While poisoning attacks on machine learning models have been extensively studied, the mechanisms by which adversaries can distribute poisoned models at scale remain largely unexplored. In this paper, we shed light on how model leaderboards -- ranked platforms for model discovery and evaluation -- can serve as a powerful channel for adversaries for stealthy large-scale distribution of poisoned models. We present TrojanClimb, a general framework that enables injection of malicious behaviors while maintaining competitive leaderboard performance. We demonstrate its effectiveness across four diverse modalities: text-embedding, text-generation, text-to-speech and text-to-image, showing that adversaries can successfully achieve high leaderboard rankings while embedding arbitrary harmful functionalities, from backdoors to bias injection. Our findings reveal a significant vulnerability in the machine learning ecosystem, highlighting the urgent need to redesign leaderboard evaluation mechanisms to detect and filter malicious (e.g., poisoned) models, while exposing broader security implications for the machine learning community regarding the risks of adopting models from unverified sources.
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.
Complex piezoelectric systems are foundational in industrial applications. Their performance, however, is challenged by the nonlinear voltage-displacement hysteretic relationships. Efficient characterization methods are, therefore, essential for reliable design, monitoring, and maintenance. Recently proposed neural operator methods serve as surrogates for system characterization but face two pressing issues: interpretability and generalizability. State-of-the-art (SOTA) neural operators are black-boxes, providing little insight into the learned operator. Additionally, generalizing them to novel voltages and predicting displacement profiles beyond the training domain is challenging, limiting their practical use. To address these limitations, this paper proposes a neuro-symbolic operator (NSO) framework that derives the analytical operators governing hysteretic relationships. NSO first learns a Fourier neural operator mapping voltage fields to displacement profiles, followed by a library-based sparse model discovery method, generating white-box parsimonious models governing the underlying hysteresis. These models enable accurate and interpretable prediction of displacement profiles across varying and out-of-distribution voltage fields, facilitating generalizability. The potential of NSO is demonstrated by accurately predicting voltage-displacement hysteresis, including butterfly-shaped relationships. Moreover, NSO predicts displacement profiles even for noisy and low-fidelity voltage data, emphasizing its robustness. The results highlight the advantages of NSO compared to SOTA neural operators and model discovery methods on several evaluation metrics. Consequently, NSO contributes to characterizing complex piezoelectric systems while improving the interpretability and generalizability of neural operators, essential for design, monitoring, maintenance, and other real-world scenarios.
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.
Machine Learning produces efficient decision and prediction models based on input-output data only. Such models have the form of decision trees or neural nets and are far from transparent analytical models, based on mathematical formulas. Analytical model discovery requires additional knowledge and may be performed with Grammatical Evolution. Such models are transparent, concise, and have readable components and structure. This paper reports on a non-trivial experiment with generating such models.
We introduce the first Automatically Differentiable Model Updating (ADiMU) framework that finds any history-dependent material model from full-field displacement and global force data (global, indirect discovery) or from strain-stress data (local, direct discovery). We show that ADiMU can update conventional (physics-based), neural network (data-driven), and hybrid material models. Moreover, this framework requires no fine-tuning of hyperparameters or additional quantities beyond those inherent to the user-selected material model architecture and optimizer. The robustness and versatility of ADiMU is extensively exemplified by updating different models spanning tens to millions of parameters, in both local and global discovery settings. Relying on fully differentiable code, the algorithmic implementation leverages vectorizing maps that enable history-dependent automatic differentiation via efficient batched execution of shared computation graphs. This contribution also aims to facilitate the integration, evaluation and application of future material model architectures by openly supporting the research community. Therefore, ADiMU is released as an open-source computational tool, integrated into a carefully designed and documented software named HookeAI.
Many model selection algorithms rely on sparse dictionary learning to provide interpretable and physics-based governing equations. The optimization algorithms typically use a hard thresholding process to enforce sparse activations in the model coefficients by removing library elements from consideration. By introducing an annealing scheme that reactivates a fraction of the removed terms with a cooling schedule, we are able to improve the performance of these sparse learning algorithms. We concentrate on two approaches to the optimization, SINDy, and an alternative using hard thresholding pursuit. We see in both cases that annealing can improve model accuracy. The effectiveness of annealing is demonstrated through comparisons on several nonlinear systems pulled from convective flows, excitable systems, and population dynamics. Finally we apply these algorithms to experimental data for projectile motion.
A fundamental challenge in the cognitive sciences is discovering the dynamics that govern behaviour. Take the example of spoken language, which is characterised by a highly variable and complex set of physical movements that map onto the small set of cognitive units that comprise language. What are the fundamental dynamical principles behind the movements that structure speech production? In this study, we discover models in the form of symbolic equations that govern articulatory gestures during speech. A sparse symbolic regression algorithm is used to discover models from kinematic data on the tongue and lips. We explore these candidate models using analytical techniques and numerical simulations, and find that a second-order linear model achieves high levels of accuracy, but a nonlinear force is required to properly model articulatory dynamics in approximately one third of cases. This supports the proposal that an autonomous, nonlinear, second-order differential equation is a viable dynamical law for articulatory gestures in speech. We conclude by identifying future opportunities and obstacles in data-driven model discovery and outline prospects for discovering the dynamical principles that govern language, brain and behaviour.
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.
This study proposes a coordinated ramp metering control framework in large networks based on scalable nonlinear traffic dynamics model discovery. Existing coordinated ramp metering control methods often require accurate traffic dynamics models in real time, however, for large-scale highway networks, since these models are always nonlinear, they are extremely challenging to obtain. To overcome this limitation, this study utilizes the Sparse Identification of Nonlinear Dynamics with Control (SINDYc) to derive the accurate nonlinear traffic dynamics model from observed data. The discovered dynamics model is then integrated into a Model Predictive Control (MPC) coordinated ramp metering controller, enabling optimized control actions that enhance traffic flow and efficiency. The proposed framework is tested on a large-scale highway network that includes three intersecting highways and eight on-ramps, which outperforms the existing approaches, demonstrating its effectiveness and potential for real-time application. This framework can offer a scalable and robust solution for improving real-time traffic management in complex urban environments.
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.
This paper proposes a cutting mechanics-based machine learning (CMML) modeling method to discover governing equations of machining dynamics. The main idea of CMML design is to integrate existing physics in cutting mechanics and unknown physics in data to achieve automated model discovery, with the potential to advance machining modeling. Based on existing physics in cutting mechanics, CMML first establishes a general modeling structure governing machining dynamics, that is represented by a set of unknown differential algebraic equations. CMML can therefore achieve data-driven discovery of these unknown equations through effective cutting mechanics-based nonlinear learning function space design and discrete optimization-based learning algorithm. Experimentally verified time domain simulation of milling is used to validate the proposed modeling method. Numerical results show CMML can discover the exact milling dynamics models with process damping and edge force from noisy data. This indicates that CMML has the potential to be used for advancing machining modeling in practice with the development of effective metrology systems.
While many physical and engineering processes are most effectively described by non-linear symbolic models, existing non-linear symbolic regression (SR) methods are restricted to a limited set of continuous algebraic functions, thereby limiting their applicability to discover higher order non-linear differential relations. In this work, we introduce the Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO) method, a novel approach to symbolic model discovery that leverages Neural Operators to encompass a broad range of symbolic operations. We demonstrate that NOMTO can successfully identify symbolic expressions containing elementary functions with singularities, special functions, and derivatives. Additionally, our experiments demonstrate that NOMTO can accurately rediscover second-order non-linear partial differential equations. By broadening the set of symbolic operations available for discovery, NOMTO significantly advances the capabilities of existing SR methods. It provides a powerful and flexible tool for model discovery, capable of capturing complex relations in a variety of physical systems.
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.
Soft biological tissues exhibit a tendency to maintain a preferred state of tensile stress, known as tensional homeostasis, which is restored even after external mechanical stimuli. This macroscopic behavior can be described using the theory of kinematic growth, where the deformation gradient is multiplicatively decomposed into an elastic part and a part related to growth and remodeling. Recently, the concept of homeostatic surfaces was introduced to define the state of homeostasis and the evolution equations for inelastic deformations. However, identifying the optimal model and material parameters to accurately capture the macroscopic behavior of inelastic materials can only be accomplished with significant expertise, is often time-consuming, and prone to error, regardless of the specific inelastic phenomenon. To address this challenge, built-in physics machine learning algorithms offer significant potential. In this work, we extend our inelastic Constitutive Artificial Neural Networks (iCANNs) by incorporating kinematic growth and homeostatic surfaces to discover the scalar model equations, namely the Helmholtz free energy and the pseudo potential. The latter describes the state of homeostasis in a smeared sense. We evaluate the ability of the proposed network to learn from experimentally obtained tissue equivalent data at the material point level, assess its predictive accuracy beyond the training regime, and discuss its current limitations when applied at the structural level. Our source code, data, examples, and an implementation of the corresponding material subroutine are made accessible to the public at https://doi.org/10.5281/zenodo.13946282.
Symbolic regression (SR) methods have been extensively investigated to explore explicit algebraic Reynolds stress models (EARSM) for turbulence closure of Reynolds-averaged Navier-Stokes (RANS) equations. The deduced EARSM can be readily implemented in existing computational fluid dynamic (CFD) codes and promotes the identification of physically interpretable turbulence models. The existing SR methods, such as genetic programming, sparse regression, or artificial neural networks, require user-defined functional operators, a library of candidates, or complex optimization algorithms. In this work, a novel framework using LLMs to automatically discover algebraic expressions for correcting the RSM is proposed. The direct observation of Reynolds stress and the indirect output of the CFD simulation are both involved in the training process to guarantee data consistency and avoid numerical stiffness. Constraints of functional complexity and convergence are supplementally imposed in the objective function on account of the tremendous flexibility of LLMs. The evolutionary search is employed for global optimization. The proposed method is performed for separated flow over periodic hills at Re = 10,595. The generalizability of the discovered model is verified on a set of 2D turbulent separated flow configurations with different Reynolds numbers and geometries. It is demonstrated that the corrective RANS can improve the prediction for both the Reynolds stress and mean velocity fields. Compared with algebraic models discovered by other works, the discovered model performs better in accuracy and generalization capability. The proposed approach provides a promising paradigm for using LLMs to improve turbulence modeling for a given class of flows.
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.