Dimensionality reduction algorithms like principal component analysis (PCA) are workhorses of machine learning and neuroscience, but each has well-known limitations. Variants of PCA are simple and interpretable, but not flexible enough to capture nonlinear data manifold structure. More flexible approaches have other problems: autoencoders are generally difficult to interpret, and graph-embedding-based methods can produce pathological distortions in manifold geometry. Motivated by these shortcomings, we propose a variational framework that casts dimensionality reduction algorithms as solutions to an optimal manifold embedding problem. By construction, this framework permits nonlinear embeddings, allowing its solutions to be more flexible than PCA. Moreover, the variational nature of the framework has useful consequences for interpretability: each solution satisfies a set of partial differential equations, and can be shown to reflect symmetries of the embedding objective. We discuss these features in detail and show that solutions can be analytically characterized in some cases. Interestingly, one special case exactly recovers PCA.

The ELBO is derived from Jensen's inequality as follows: log p ( Y) ZZZ q ( X, f, w) log p ( Y, X, f, w) q ( X, f, w) d w d f d X (31) = ZZZ p ( f | w) q ( w) The inference procedure of SSGP is shown in Algorithm 1. In the experiments, we set the integration time window =1 . Update the parameters by maximizing the ELBO (13) evaluated using D . In this appendix, we describe baseline models for the experiments in Section 6. D-SymODEN can also apply to the dissipative systems. SympGPR can estimate conservative vector fields from derivative observations by considering Hamiltonian mechanics; we used finite differences for training.

We propose a Kolmogorov-Arnold Representation-based Hamiltonian Neural Network (KAR-HNN) that replaces the Multilayer Perceptrons (MLPs) with univariate transformations. While Hamiltonian Neural Networks (HNNs) ensure energy conservation by learning Hamiltonian functions directly from data, existing implementations, often relying on MLPs, cause hypersensitivity to the hyperparameters while exploring complex energy landscapes. Our approach exploits the localized function approximations to better capture high-frequency and multi-scale dynamics, reducing energy drift and improving long-term predictive stability. The networks preserve the symplectic form of Hamiltonian systems, and thus maintain interpretability and physical consistency. After assessing KAR-HNN on four benchmark problems including spring-mass, simple pendulum, two- and three-body problem, we foresee its effectiveness for accurate and stable modeling of realistic physical processes often at high dimensions and with few known parameters.

We take a physics-based approach to studying how the internal hidden states of large language models change from token to token during inference. Across 20 open-source transformer models (135M-3B parameters), we find that a quantity combining the rate of change in hidden states and the model's next-token certainty, analogous to energy in physics, remains nearly constant. Random-weight models conserve this "energy" more tightly than pre-trained ones, while training shifts models into a faster, more decisive regime with greater variability. Using this "log-Lagrangian" view, we derive a control method called Jacobian steering, which perturbs hidden states in the minimal way needed to favor a target token. This approach maintained near-constant energy in two tested models and produced continuations rated higher in semantic quality than the models' natural outputs. Viewing transformers through this mechanics lens offers a principled basis for interpretability, anomaly detection, and low-risk steering. This could help make powerful models more predictable and aligned with human intent.

Laboratory of Computational Science and Modeling, Institut des Mat eriaux, Ecole Polytechnique F ed erale de Lausanne, 1015 Lausanne, Switzerland (Dated: August 5, 2025) The equations of classical mechanics can be used to model the time evolution of countless physical systems, from the astrophysical to the atomic scale. Accurate numerical integration requires small time steps, which limits the computational efficiency - especially in cases such as molecular dynamics that span wildly different time scales. Using machine-learning (ML) algorithms to predict trajectories allows one to greatly extend the integration time step, at the cost of introducing artifacts such as lack of energy conservation and loss of equipartition between different degrees of freedom of a system. We propose learning data-driven structure-preserving (symplectic and time-reversible) maps to generate long-time-step classical dynamics, showing that this method is equivalent to learning the mechanical action of the system of interest. We show that an action-derived ML integrator eliminates the pathological behavior of non-structure-preserving ML predictors, and that the method can be applied iteratively, serving as a correction to computationally cheaper direct predictors. Simulating classical mechanical systems with high accuracy and efficiency is a long-standing challenge in computational physics [31, 32]. Traditional numerical methods typically rely on small time steps to propagate the equations of motion for the dynamical system in order to provide accurate integration.

The increasing amount of pressure related to water and energy shortages has increased the urgency of cultivating individual conservation behaviors. While the concept of nudging, i.e., providing usage-based feedback, has shown promise in encouraging conservation behaviors, its efficacy is often constrained by the lack of targeted and actionable content. This study investigates the impact of the use of large language models (LLMs) to provide tailored conservation suggestions for conservation intentions and their rationale. Through a randomized controlled trial with 1,515 university participants, we compare three virtual nudging scenarios: no nudging, traditional nudging with usage statistics, and LLM-powered nudging with usage statistics and personalized conservation suggestions. The results of statistical analyses and causal forest modeling reveal that nudging led to an increase in conservation intentions among 86.9%-98.0% of the participants. LLM-powered nudging achieved a maximum increase of 18.0% in conservation intentions, surpassing traditional nudging by 88.6%. Furthermore, structural equation modeling results reveal that exposure to LLM-powered nudges enhances self-efficacy and outcome expectations while diminishing dependence on social norms, thereby increasing intrinsic motivation to conserve.

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

Data-driven machine learning models often require extensive datasets, which can be costly or inaccessible, and their predictions may fail to comply with established physical laws. Current approaches for incorporating physical priors mitigate these issues by penalizing deviations from known physical laws, as in physics-informed neural networks, or by designing architectures that automatically satisfy specific invariants. However, penalization approaches do not guarantee compliance with physical constraints for unseen inputs, and invariant-based methods lack flexibility and generality. We propose a novel physics-consistent machine learning method that directly enforces compliance with physical principles by projecting model outputs onto the manifold defined by these laws. This procedure ensures that predictions inherently adhere to the chosen physical constraints, improving reliability and interpretability. Our method is demonstrated on two systems: a spring-mass system and a low-temperature reactive plasma. Compared to purely data-driven models, our approach significantly reduces errors in physical law compliance, enhances predictive accuracy of physical quantities, and outperforms alternatives when working with simpler models or limited datasets. The proposed projection-based technique is versatile and can function independently or in conjunction with existing physics-informed neural networks, offering a powerful, general, and scalable solution for developing fast and reliable surrogate models of complex physical systems, particularly in resource-constrained scenarios.

Most current neural networks for molecular dynamics (MD) include physical inductive biases, resulting in specialized and complex architectures. This is in contrast to most other machine learning domains, where specialist approaches are increasingly replaced by general-purpose architectures trained on vast datasets. In line with this trend, several recent studies have questioned the necessity of architectural features commonly found in MD models, such as built-in rotational equivariance or energy conservation. In this work, we contribute to the ongoing discussion by evaluating the performance of an MD model with as few specialized architectural features as possible. We present a recipe for MD using an Edge Transformer, an "off-the-shelf'' transformer architecture that has been minimally modified for the MD domain, termed MD-ET. Our model implements neither built-in equivariance nor energy conservation. We use a simple supervised pre-training scheme on $\sim$30 million molecular structures from the QCML database. Using this "off-the-shelf'' approach, we show state-of-the-art results on several benchmarks after fine-tuning for a small number of steps. Additionally, we examine the effects of being only approximately equivariant and energy conserving for MD simulations, proposing a novel method for distinguishing the errors resulting from non-equivariance from other sources of inaccuracies like numerical rounding errors. While our model exhibits runaway energy increases on larger structures, we show approximately energy-conserving NVE simulations for a range of small structures.