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1. A hierarchy of thermodynamics learning frameworks for inelastic constitutive modeling

   Jones, RE; Fuhg, JN (March 2026, arXiv: https://arxiv.org/pdf/2603.02645)

   Recent advances in physics-augmented neural networks have enabled thermodynamically consistent data-driven constitutive modeling of complex inelastic materials. Most existing approaches, however, implicitly adopt a specific thermodynamic framework and embed structural assumptions such as normality, dual dissipation potentials, or other structure from manually constructed models directly into the learning architecture. Consequently, differences in predictive performance may arise not only from data or network design, but also from the underlying theoretical assumptions. In this work, we present a unified comparison of several thermodynamically consistent inelastic modeling frameworks from a machine learning perspective. We consider internal-variable formulations with dissipation potential, generalized standard materials, and metriplectic structures, and we analyze their structural assumptions, admissible dependencies, convexity requirements, and implications for dissipation and evolution. Each framework is implemented within a common neural potential architecture based on invariant representations and neural ordinary differential equations. This unified setting ensures that performance differences can be attributed to thermodynamic structure rather than architectural variation. The models are trained and evaluated on three representative inelastic datasets generated from high-fidelity representative volume element simulations: an elastoplastic alloy, a viscoelastic composite, and a rate-dependent crystal plasticity polycrystal. By isolating the role of thermodynamic structure, we assess how restrictions such as duality, normality, operator-based evolution, and convexity influence learnability, expressiveness, stability, and generalization. 

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2. Symphony in the Latent Space: Provably Integrating High-dimensional Techniques with Non-linear Machine Learning Models

   Qiong Wu; Jian Li; Zhenming Liu; Yanhua Li; Mihai Cucuringu (April 2023, Proceedings of the AAAI Conference on Artificial Intelligence)

   This paper revisits building machine learning algorithms that involve interactions between entities, such as those between financial assets in an actively managed portfolio, or interac- tions between users in a social network. Our goal is to forecast the future evolution of ensembles of multivariate time series in such applications (e.g., the future return of a financial asset or the future popularity of a Twitter account). Designing ML algorithms for such systems requires addressing the challenges of high-dimensional interactions and non-linearity. Existing approaches usually adopt an ad-hoc approach to integrating high-dimensional techniques into non-linear models and re- cent studies have shown these approaches have questionable efficacy in time-evolving interacting systems. To this end, we propose a novel framework, which we dub as the additive influence model. Under our modeling assump- tion, we show that it is possible to decouple the learning of high-dimensional interactions from the learning of non-linear feature interactions. To learn the high-dimensional interac- tions, we leverage kernel-based techniques, with provable guarantees, to embed the entities in a low-dimensional latent space. To learn the non-linear feature-response interactions, we generalize prominent machine learning techniques, includ- ing designing a new statistically sound non-parametric method and an ensemble learning algorithm optimized for vector re- gressions. Extensive experiments on two common applica- tions demonstrate that our new algorithms deliver significantly stronger forecasting power compared to standard and recently proposed methods. 

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3. Symphony in the Latent Space: Provably Integrating High-dimensional Techniques with Non-linear Machine Learning Models

   https://doi.org/10.1609/aaai.v37i9.26233  

   Wu, Qiong; Li, Jian; Liu, Zhenming; Li, Yanhua; Cucuringu, Mihai (February 2023, AAAI)

   This paper revisits building machine learning algorithms that involve interactions between entities, such as those between financial assets in an actively managed portfolio, or interactions between users in a social network. Our goal is to forecast the future evolution of ensembles of multivariate time series in such applications (e.g., the future return of a financial asset or the future popularity of a Twitter account). Designing ML algorithms for such systems requires addressing the challenges of high-dimensional interactions and non-linearity. Existing approaches usually adopt an ad-hoc approach to integrating high-dimensional techniques into non-linear models and re- cent studies have shown these approaches have questionable efficacy in time-evolving interacting systems. To this end, we propose a novel framework, which we dub as the additive influence model. Under our modeling assump- tion, we show that it is possible to decouple the learning of high-dimensional interactions from the learning of non-linear feature interactions. To learn the high-dimensional interac- tions, we leverage kernel-based techniques, with provable guarantees, to embed the entities in a low-dimensional latent space. To learn the non-linear feature-response interactions, we generalize prominent machine learning techniques, includ- ing designing a new statistically sound non-parametric method and an ensemble learning algorithm optimized for vector re- gressions. Extensive experiments on two common applica- tions demonstrate that our new algorithms deliver significantly stronger forecasting power compared to standard and recently proposed methods. 

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4. Deep Spatiotemporal Point Processes: Advances and New Directions

   https://doi.org/10.1146/annurev-statistics-042324-040052  

   Cheng, Xiuyuan; Dong, Zheng; Xie, Yao (March 2026, Annual Review of Statistics and Its Application)

   Spatiotemporal point processes model discrete events distributed in space and time, with applications in criminology, seismology, epidemiology, and social networks. Classical models rely on parametric kernels, limiting their ability to capture heterogeneous, nonstationary dynamics. Recent advances integrate deep neural architectures, either by modeling the conditional intensity directly or by learning flexible, data-driven influence kernels. This article reviews the deep influence kernel approach, which balances statistical interpretability by retaining explicit kernels to capture event propagation, with expressive power from neural architectures. We outline key components, including functional basis decomposition, graph neural networks for encoding spatial or network structures, and both likelihood-based and likelihood-free estimation methods, while addressing scalability for large data. We also highlight theoretical results on kernel identifiability. Applications in crime analysis, earthquake aftershock prediction, and sepsis modeling demonstrate the framework's effectiveness. We conclude with promising directions for developing explainable and scalable deep kernel point processes. 

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5. Microstructure-Sensitive Deformation Modeling and Materials Design with Physics-Informed Neural Networks

   https://doi.org/10.2514/1.J062708  

   Hasan, Mahmudul; Ender_Eger, Zekeriya; Senthilnathan, Arulmurugan; Acar, Pınar (May 2024, AIAA Journal)

   Microstructure-sensitive materials design has become popular among materials engineering researchers in the last decade because it allows the control of material performance through the design of microstructures. In this study, the microstructure is defined by an orientation distribution function. A physics-informed machine learning approach is integrated into microstructure design to improve the accuracy, computational efficiency, and explainability of microstructure-sensitive design. When data generation is costly and numerical models need to follow certain physical laws, machine learning models that are domain-aware perform more efficiently than conventional machine learning models. Therefore, a new paradigm called the physics-informed neural network (PINN) is introduced in the literature. This study applies the PINN to microstructure-sensitive modeling and inverse design to explore the material behavior under deformation processing. In particular, we demonstrate the application of PINN to small-data problems driven by a crystal plasticity model that needs to satisfy the physics-based design constraints of the microstructural orientation space. For the first problem, we predict the microstructural texture evolution of copper during a tensile deformation process as a function of initial texturing and strain rate. The second problem aims to calibrate the crystal plasticity parameters of the Ti-7Al alloy by solving an inverse design problem to match the PINN-predicted final texture prediction and the experimental data. 

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