We develop a variational and geometric theory based on statistical physics for the joint evolution of Transformer attention and token representations at fixed model parameters. To this end, we introduce a free energy for the attention and model the evolution of the attention weights by its Fisher--Rao gradient flow. At fixed token representations, we prove that the softmax matrix is the unique free-energy minimiser and that every positive attention trajectory converges to it. For prescribed time-dependent representations, we show that the attention dynamics becomes linear after a change to logit variables. This result yields tracking bounds relative to the instantaneous softmax and sufficient conditions for attention to converge to a fixed matrix. Hamiltonian dynamics for the representations and their momenta are then coupled to the attention relaxation driven by the same energy. This coupling results in an energy-dissipation law. We also develop two alternative Hamiltonians incorporating the attention value map. We prove global well-posedness of the attention flow and all three coupled systems.
Reinforcement learning (RL) solves the explore-exploit problem by training a learner on reward; its multi-agent, resource-sharing form has an equilibrium allocation as its optimum -- the ideal-free distribution (IFD), the Nash equilibrium of a congestion game. We show a minimal swarm approximates it with no learner at all. We present a decentralized, learner-free method for this problem: agents explore by heavy-tailed (Levy) flights, recruit neighbours to a discovery through a degree-normalized threshold cascade, and disperse as a patch crowds. The method drives the collective to the shape of the equilibrium allocation (pooled spatial matching slope 1.14, R^2=0.61) with no training, no stored parameters, and no reward in its dynamics; it reaches this as a self-organized attractor and holds it, re-finding it when the resources move -- though it reaches the shape, not the exact equilibrium, and stays measurably exploitable (Nash-gap 1.46). Coupling recruitment to movement causes the allocation: removing it collapses the matching (slope 0.006). Against Proximal Policy Optimization (PPO) learners (feedforward, centralized-critic, graph message-passing, and recurrent) trained in the same sensing-limited environment, the method at zero training is the only arm left unbiased w.r.t. the IFD gradient (slope ~1, CI containing 1, vs learners 0.69-0.97, all under-concentrating) and ties the best learner's harvest where resource is clustered, while the learner collects more where resource is scarce (0.33 vs 0.22) and holds a tighter per-capita intake. Learning buys precision and transfer across reward structures, not a better allocation shape. A companion paper develops the mechanism.
A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $\Sigma\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $\Omega(n^3)$ copies, strictly exceeding the sample complexity $\Theta(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $\Sigma\ge(\frac12+\nu)I$ for any parameter $\nu>0$, we prove that single-copy tomography requires $N=\Theta\left(n^2\min(n,1+\nu^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $\nu=\Omega(1)$, the sample complexity drops to $\Theta(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.
Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap, $\Delta$, with an $O(\Delta)$ gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. Finally, we show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.
Understanding how (multimodal) large language models perform on physics problems requires benchmarks that reflect the difficulty and breadth of expert-level physical reasoning. Existing physics benchmarks remain limited in the following two important ways: (1) short of high-difficulty datasets, and (2) lack of comprehensive coverage of visual forms, knowledge points, and step-by-step solution processes. As a result, model performance on current datasets may not be fully representative of their ability to solve complex physics problems. To address these issues, we present PhysElite, a large-scale bilingual multimodal benchmark for Olympiad-level physics reasoning. PhysElite contains 11,586 Olympiad-tier problems. For each problem, we provide corresponding visual diagrams, step-by-step bilingual Chinese-English solution derivations, and the final answer. We benchmark 18 open-source and closed-source MLLMs, and find that even the strongest model reaches only 33.7% answer accuracy. We additionally conduct step-level process evaluation to diagnose where models fail in the reasoning chain. Our datasets are released at https://github.com/PremiLab-Math/PhysElite.
Symmetry governs the organization of physical structures, while symmetry breaking reveals their transformations. In geometric deep learning, symmetry imposes rank constraints on equivariant representations, limiting the expressive capacity of equivariant graph neural networks (GNNs). Here we show that these constraints can instead be exploited: the singular-value spectra of equivariant representations provide a unified fingerprint of structural symmetry. We introduce spectral symmetry fingerprints (SymFP), which enable discrete point-group identification through rank-template matching and continuous symmetry analysis by quantifying spectral energy beyond target rank constraints. SymFP is orientation invariant and architecture agnostic, enabling a parameter-free GNN implementation. Its spectra can be reused across multiple target groups, and the resulting leakage scores satisfy subgroup monotonicity. Across synthetic structures, molecules, coordination complexes and large biological assemblies, SymFP accurately identifies point groups, tracks controlled symmetry breaking and reveals local and degree-resolved symmetry changes that can be obscured by global geometric measures. These results establish the spectral structure of equivariant representations as a general framework for characterizing symmetry across physical systems.
Data-driven methods can recover compact equations, yet the physical dependence of their coefficients across operating conditions often remains unresolved. Dimensionless groups can explain this variation, but selecting relevant groups from the many dimensionally admissible combinations remains challenging. Here, we introduce a knowledge-guided framework for dimensionless learning, called {\pi}-former. It learns patterns of variable association and exponent assignment from a corpus of 1,288 established dimensionless groups. Under exact dimensional constraints, the model proposes interpretable basic groups whose combinations are optimized against observations. Across seven physical problems, the framework identifies predictive dimensionless coordinates within seconds. Integration with equation discovery further expresses condition-dependent coefficients through these coordinates, recovering established physical scalings in multi-condition simulations. Applied to experimental measurements of evaporation-driven wetting, the framework discovers dimensionless coordinates and uses them to construct a unified trajectory equation. The resulting model generalizes to held-out conditions across liquids and initial volumes, outperforming existing empirical models. These results establish dimensionless learning as a transferable bridge between accumulated scientific knowledge and data-driven equation discovery, enabling artificial intelligence to uncover physically meaningful laws from observations.
Uncertainty quantification (UQ) for random partial differential equations (PDEs) is ubiquitous in computational science and engineering. However, classical spectral solvers for this class of problems face the curse of dimensionality, and existing neural solvers often ignore the stochastic structure that makes moments and calibration tractable. We introduce a stochastic separable physics-informed neural network, dubbed S$^{2}$-PINN, that represents the solution $u(t,\mathbf{x},\mathbf{Z})$ of a random PDE with a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos (gPC) stochastic basis, coupled by a low-rank Canonical Polyadic (CP) tensor decomposition core. The method is trained with a hybrid strong-form and gPC-projected residual loss. Our theoretical analysis establishes that the separable class is dense in $L^2$ under mild conditions, and the projected residual corresponds exactly to a stochastic Galerkin constraint. Furthermore, we show that mini-batch projection coefficients are logarithmically dependent on the number of gPC modes, and that the orthogonality penalty controls the conditioning of the learned spatial dictionary. Using four manufactured random PDE benchmarks, we show that S$^{2}$-PINN outperforms nine baselines in terms of mean and variance accuracy, as well as calibration, while using significantly fewer parameters. Further evaluations on non-manufactured Poisson and Darcy problems, a stochastic Navier--Stokes problem, a diffusion scaling study of higher random dimensions, and two stochastic inverse problems reveal the generalization capabilities of the proposed structure. Together, these results support stochastic separability as an effective design principle for physics-informed neural UQ. The code for the experiments can be found in https://github.com/DMax1314/s2pinn
The Lane-Emden Equation is a class of nonlinear differential equations commonly used to model phenomena in astrophysics and mathematical physics, such as stellar structure and thermal behavior in polytropic gas spheres. This study presents a novel hybrid framework that integrates Physics-Informed Neural Networks (PINNs) with boosting methods to enhance the accuracy and robustness of solutions for Lane-Emden type differential equations. Standard PINNs leverage physical laws encoded within the loss function, but their predictive performance can be constrained by optimization challenges and approximation errors. To address this, we introduce a post-processing boosting stage where the residual errors from the PINN prediction are modeled using some regression algorithms - including LightGBM, Random Forest, XGBoost, and Support Vector Regression (SVR). The outputs from these boosted models are aggregated to generate a corrected, high-fidelity prediction. The proposed methodology is systematically validated through quantitative metrics and comparative analysis against the standard PINN baseline. The results demonstrate that the boosted PINN framework significantly reduces prediction error, and improves generalization, offering a powerful and flexible approach for solving scientific and engineering problems governed by differential equations.
Transient pressure diffusion in heterogeneous porous media becomes difficult to resolve efficiently when permeability is discontinuous and spans several orders of magnitude. In this paper, we develop a domain-decomposed random-feature evolutional deep neural network (DD RF-EDNN) that separates spatial approximation from temporal evolution. Permeability-informed random features are compressed into an orthonormal pressure space, and a conservative finite-volume operator is projected onto this space so that only the reduced coordinates are advanced in time. This formulation retains the dissipative structure of the discrete flow problem without repeated neural-network optimization. The analysis quantifies dictionary and singular-value truncation errors, proves contractivity under an energy-consistent dissipativity correction, and derives a conditional grid-level estimate for homogeneous dynamics and time-independent data admitting a steady lifting. Numerical experiments on low-permeability inclusions, high-conductivity blocks and channels, and three-dimensional grids assess the accuracy of the method across distinct coefficient configurations. The method reaches a final-time relative $L^2$ error of $9.54\times10^{-4}$ in the principal benchmark and maintains errors on the order of $10^{-2}$ at a permeability contrast of $10^3$. Numerical diagnostics further show that the observed error balance and temporal convergence are consistent with the analysis. These results support DD RF-EDNN as a structure-preserving and interpretable reduced formulation for transient simulations on prescribed heterogeneous media.
While universal approximation theorems theoretically underpin the representation capabilities of machine learning models, they have two shortcomings: They do not focus on efficient representations and they tend to be insensitive to depth and other heuristically important properties. In this work we show that these two shortcomings are intertwined. We prove that machine learning models which are deeper or have higher-order interactions can efficiently represent a larger set of functions, i.e. have larger efficient reach. It follows that deep Boltzmann machines (DBMs) have maximal efficient reach, restricted Boltzmann machines (RBMs) sub-maximal, and k-body generalizations of RBMs form a hierarchy between the two, even though all are universal. More generally, (i) we provide a full characterization of machine learning models with maximal efficient reach; (ii) we derive separations between deep and shallow models for three representation tasks; and (iii) we prove that the efficient reach of shallow models constitutes an infinite hierarchy indexed by the interaction order. The proofs draw on different methodologies, namely on (i) a recent characterization of universal spin models; (ii) computational complexity bounds for spin models; and (iii) communication complexity bounds applied to shallow models. This work shows that universal machine learning models can strongly differ in their efficient representation capabilities, and provides a theoretical foundation for heuristic observations.
Hybrid domain decomposition methods provide a flexible framework for coupling full order models (FOMs) and reduced order models (ROMs), but typically assume the model assigned to each subdomain is fixed throughout a simulation. This is limiting for transient problems in which localized features propagate through the domain and the regions requiring high-fidelity resolution change over time. We introduce a reinforcement learning (RL)-based approach for online adaptation of FOM-ROM models coupled via the overlapping Schwarz alternating method (O-SAM), an iterative domain decomposition method that solves subdomain-local problems while exchanging solution information through transmission boundary conditions on overlapping interfaces. Deep Q-networks (DQNs) are trained offline to select among subdomain-local FOMs and pre-trained Operator Inference (OpInf) ROMs using a reward balancing accuracy, cost, and model-switching frequency. Once trained, the policies are deployed predictively on problem instances not seen during training, without requiring a reference FOM solution. We demonstrate the approach on two examples: a 1D advection-diffusion problem with a moving front, and a 3D linear elastic wave propagation problem implemented in the Norma.jl solid mechanics code. For the advection-diffusion benchmark, the learned policy dynamically allocates high-fidelity resolution as the front propagates and outperforms static FOM/ROM assignments; letting the agent also adapt the domain decomposition provides no further benefit. For the elastic wave benchmark, learned policies for two and three subdomain decompositions track the propagating wave by assigning FOMs to subdomains containing the wave and ROMs elsewhere, as expected. Our results demonstrate the potential of RL to enable predictive online adaptation of model fidelity within Schwarz-based hybrid simulations.