arXiv subjects

Qianxiao Li

Publications and source records attributed to Qianxiao Li.

At least 19 recordsLinked to original sources

arXiv2026

Hidden Phase Transition in Single-Polymer Stretching under Elongational Flow

Using machine learning, we identify a second transition in polymer chain stretching driven by planar elongational flow, beyond the well-known coil-stretch transition. We achieve this by developing a conditional extension of Universal and Identifiable Dissipative Dynamics (UIDD) that jointly learns reduced stochastic dynamics from microscopic trajectories across flow strengths and yields the corresponding stationary nonequilibrium free energy landscapes. As the flow strength increases, the evolving landscape exhibits the expected coil-stretch transition, and then a previously unrecognized dynamical transition. In the latter, the stretched states persist, but the distribution along a folding coordinate becomes bimodal when conditioned on small end-to-end separation. Large-deviation analysis of a simplified analytical chain model reproduces both transitions and shows that they are phase transitions in the long-chain limit. Single-molecule DNA experiments provide qualitative support for the dynamical transition through the predicted reversal of the folding response.

cond-mat.soft↗

arXiv2026

Deep Mesoscopic Dynamics

Microscopic simulations resolve spatial patterns but are feasible only over small domains and short timescales, whereas bulk descriptions erase those patterns through spatial averaging. Mesoscopic fields preserve this spatial organisation, but their effective equations are known only for selected microscopic systems and scaling limits. We introduce Deep Mesoscopic Dynamics (DeepMesoD), which uses neural networks to learn an effective mesoscopic field equation from microscopic trajectories at a prescribed coarse-graining scale. For the nonconserved dissipative dynamics studied here, DeepMesoD learns the free-energy functional with a spatial interaction kernel and a state-dependent dissipation operator constrained by the fluctuation-dissipation relation. We first validate the construction in the Ising-Kac model, where the learned mesoscopic equation recovers the known hydrodynamic limit and rollouts remain stable far beyond the training horizon. We then treat two systems without closed-form mesoscopic descriptions. For a two-dimensional Heisenberg ferromagnet governed by atomistic spin dynamics, DeepMesoD learns a mesoscopic model whose predictions match atomistic references up to 1 ns, and a rollout with an unseen initial condition remains bounded up to 25 ns. For layered CrBr$_3$, the learned kernel captures the suppression and sign reversal of interlayer coupling at a stacking fault whose location and strength are not supplied as inputs. Controlled changes to the learned interaction show how fault number and placement shape multistep magnetic responses, complementing our magneto-photoluminescence measurements. Across the two atomistic systems, DeepMesoD achieves a three-to-four-order-of-magnitude reduction in wall-clock time compared with atomistic spin dynamics.

cs.LG↗

arXiv2026

Learning Macroscopic Dynamics without Reconstructing Microscopic States

Modeling the temporal evolution of macroscopic properties of complex systems is an important scientific task. To predict this evolution without full microscopic simulation, a common approach encodes microstates into compact latent states, learns their evolution, and reads out macroscopic predictions from the latent trajectory. These latent states are often learned through microstate reconstruction. However, with limited latent capacity, reconstruction can favor high-variance microscopic details over information needed for macroscopic prediction. Yet jointly learning latent states and their transition without reconstruction often fails to obtain latent dynamics that support accurate macroscopic prediction. We show that this failure can arise from latent scale collapse: shrinking the latent state scale reduces training loss while macroscopic evolution error remains large. Here, we propose a reconstruction-free framework to learn latent states with their dynamics for prescribed macroscopic prediction. Training alternates between updating the latent representation with the transition and next-state latent targets fixed, and updating the transition with the latent representation fixed. At inference, the trained model predicts macroscopic states recursively from an initial microstate. Our theoretical analysis characterizes reconstruction misalignment and scale collapse under joint training, and gives a sufficient condition for local convergence to correct latent dynamics for our method. Experiments on epidemic spreading on a lattice, mixing of two particle species, and polymer stretching demonstrate that the proposed method achieves substantially better macroscopic prediction over baselines.

cs.LG↗

arXiv2026

Learning ab initio phase-field models

Simulating microstructure evolution requires quantum-mechanical accuracy and mesoscopic reach in length and time scales, a combination that no current method achieves. Classical phase-field models provide this reach, but their accuracy is limited by phenomenological free energies and mobilities. Here we develop a framework for learning ab initio phase-field models, where the mesoscopic equation is not postulated but derived from a Mori-Zwanzig projection of molecular dynamics onto species-density fields under explicit assumptions. The nonlocal free energy and mobility left unspecified by this equation are parametrized by neural networks and learned from short molecular dynamics trajectories generated with machine-learning interatomic potentials of ab initio accuracy. We demonstrate the framework on an iron-boron melt and on hydrogen-helium mixtures under planetary conditions. For iron-boron, the model shows that the melt at the FeB$_4$ composition is spinodally unstable at ambient pressure but stabilized at 10 GPa, offering a thermodynamic rationale for why FeB$_4$ has been synthesized only under high pressure. For hydrogen-helium, the model predicts the immiscibility boundary and captures droplet nucleation and growth in helium-rain simulations of a column corresponding to 2.2 million atoms, far beyond the scale of atomistic modeling at comparable accuracy. Trained across compositions and conditions, such models could provide a mesoscopic counterpart to ab initio molecular dynamics.

cond-mat.stat-mech↗

arXiv2026

Interfacial melting as a thermodynamic indicator of solid-state synthesizability

Computational materials discovery commonly ranks candidate materials by their thermodynamic stability on the formation energy convex hull, yet many predicted-stable phases resist synthesis. We propose that solid-state synthesizability through interfacial-melt-mediated routes requires an additional thermodynamic condition: the interfacial melt at the target composition must itself remain locally stable against spinodal decomposition. We examine this in the classical Fe--B system, where thermodynamically stable FeB$_4$ has been reported under high-pressure synthesis but not in low-pressure synthesis attempts. Using melt--quench molecular dynamics driven by a fine-tuned machine-learning interatomic potential, we find that, at ambient pressure, the B-rich interfacial melt near the FeB$_4$ composition develops a concave free-energy landscape, signaling a demixing instability that is corroborated by the concentration--concentration structure factor and correlated with low-energy icosahedral and pentagonal-pyramidal boron motifs. In contrast to FeB$_4$, metastable Fe$_3$B and Fe$_{23}$B$_6$ remain synthesizable because their corresponding melts are stable. Applied pressure introduces a convex $PV$ contribution that strongly suppresses this instability, reducing the curvature at the FeB$_4$ composition to within the uncertainty of our fit at 1800~K, consistent with the experimental synthesis boundary. Comparison with CrB$_4$ further shows that weaker melt instability correlates with easier experimental synthesis. Interfacial-melt stability, which atomistic simulations can assess via the low-$k$ concentration--concentration structure factor, is thus proposed as a practical thermodynamic screening descriptor of synthesizability for AI-assisted materials discovery.

cond-mat.mtrl-sci↗

arXiv2026

Performance-Efficiency Tradeoffs in Transformers: An Approximation Theory Perspective

Transformers have achieved remarkable successes across a wide range of applications, yet the theoretical foundation of their model efficiency remains underexplored. In this work, we investigate how the model parameters -- mainly attention heads and head dimensions -- should be allocated across layers to balance expressivity and efficiency. We first provide mathematical analysis on the role of early layers in information extraction from an approximation perspective, with a theoretical characterization on the trade-off between the number of heads and head dimension under a fixed parameter budget. In addition, we uncover and prove the \emph{saturation} behavior of softmax activations: Continuously increasing head dimensions can lead to diminishing returns in learning errors, particularly for long sequences. Supported by both theory and experiments, this saturation pattern suggests that later layers can operate more efficiently with reduced parameters. Combining these insights, we propose principled strategies for allocating attention heads and dimensions across Transformers' layers, shedding light on theoretically-grounded model efficiency of Transformer-based architectures.

cs.LG↗

arXiv2026

Numerical Investigation of Sequence Modeling Theory using Controllable Memory Functions

The evolution of sequence modeling architectures, from recurrent neural networks and convolutional models to Transformers and structured state-space models, reflects ongoing efforts to address the diverse temporal dependencies inherent in sequential data. Despite this progress, systematically characterizing the strengths and limitations of these architectures remains a fundamental challenge. In this work, we propose a synthetic benchmarking framework to evaluate how effectively different sequence models capture distinct temporal structures. The core of this approach is to generate synthetic targets, each characterized by a parametric memory function $ρ(s, α)$ and a controllable parameter $α$ that determines the temporal strength. This setup allows us to produce a continuum of tasks that vary in temporal complexity, enabling fine-grained analysis of model behavior with respect to specific memory properties. We focus on four representative memory functions, each corresponding to a distinct class of temporal structures: exponential and polynomial functions for decay dynamics, impulse functions for long-range dependencies, and Airy functions for sparsity patterns. Experiments on several sequence modeling architectures confirm existing theoretical insights and reveal new findings regarding approximation capabilities, optimization dynamics, and architectural trade-offs. These results demonstrate the effectiveness of the proposed method in advancing theoretical understanding and highlight the importance of using controllable targets with clearly defined structures for evaluating sequence modeling architectures.

cs.LG↗

arXiv2026

Learning Permutation-invariant Macroscopic Dynamics

Accurately modeling the macroscopic dynamics of high-dimensional microscopic systems is of broad interest across the sciences. Many data-driven approaches learn a low-dimensional latent state through an autoencoder trained for pointwise input reconstruction. These methods typically assume a fixed ordering of microscopic degrees of freedom in the input. However, in many settings, such as particle systems, the microscopic state is inherently unordered. This motivates an autoencoder framework that learns permutation-invariant latent representations. To this end, we adopt a permutation-invariant encoder and design the decoder to reconstruct the mass distribution centered at the observed points rather than per-sample reconstruction. We then jointly learn the macroscopic dynamics of the observables together with the latent states. We demonstrate the effectiveness and robustness of the proposed method across a range of microscopic settings, including learning the energy dynamics in interacting particle systems, predicting mixing dynamics in Lennard-Jones fluids, and modeling the stretching dynamics from video data of polymers moving in an elongational force field.

cs.LG↗

arXiv2026

InfoFlow: A Framework for Multi-Layer Transformer Analysis

While the approximation properties of single-layer Transformer architectures have been studied in recent works, a rigorous theoretical understanding of the multi-layer setting remains limited. In this work, we establish that multi-layer Transformers possess fundamentally different approximation capabilities from single-layer ones: for certain retrieval tasks, any single-layer Transformer requires least $Ω(\varepsilon^{-k})$ parameters to achieve precision $\varepsilon$, where $k$ grows linearly with sequence length $T$, whereas a two-layer Transformer with a single head per layer achieves the same approximation precision with at most $O (\varepsilon^{-1})$ parameters. To understand this separation, we identify two structural mechanisms underlying multi-layer approximation. Specifically, softmax attention can only efficiently retrieve the token attaining the maximum attention score, incurring exponential-in-length parameter cost for $k$-th largest retrieval with $k \geq 2$. Moreover, the parameter cost of decoding coupled information scales with the size of the retrieved token set. Motivated by these findings, we propose InfoFlow, a framework for multi-layer Transformers. The framework tracks an information set of accessible input positions at each token and layer, assigning an explicit approximation rate to each mode of information propagation. This abstraction recovers known approximation bounds, remains consistent with experimental observations on trained networks, and yields concrete predictions in settings where direct theoretical analysis is currently intractable. Our results provide a principled framework for reasoning about the approximation efficiency of multi-layer Transformers.

cs.LG↗

arXiv2026

Hypothesis-driven construction of mesoscopic dynamics

Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. Data from each problem instance is then used to guide the identification of members within our hypothesis class, giving rise to accurate, robust and interpretable dynamical models. We empirically validate this framework on both data from continuum PDE models as a check, and on data arising from microscopic chain models for which exact meso-scale models are unknown. The proposed approach not only acts as an effective dynamics learner, but also offers vital interpretable diagnostics of the underlying physics.

cs.LG↗

arXiv2026

The Effect of Attention Head Count on Transformer Approximation

Transformer has become the dominant architecture for sequence modeling, yet a detailed understanding of how its structural parameters influence expressive power remains limited. In this work, we study the approximation properties of transformers, with particular emphasis on the role of the number of attention heads. Our analysis begins with the introduction of a generalized $D$-retrieval task, which we prove to be dense in the space of continuous functions, thereby providing the basis for our theoretical framework. We then establish both upper and lower bounds on the parameter complexity required for $ε$-approximation. Specifically, we show that transformers with sufficiently many heads admit efficient approximation, whereas with too few heads, the number of parameters must scale at least as $O(1/ε^{cT})$, for some constant $c$ and sequence length $T$. To the best of our knowledge, this constitutes the first rigorous lower bound of this type in a nonlinear and practically relevant setting. We further examine the single-head case and demonstrate that an embedding dimension of order $O(T)$ allows complete memorization of the input, where approximation is entirely achieved by the feed-forward block. Finally, we validate our theoretical findings with experiments on both synthetic data and real-world tasks, illustrating the practical relevance of our results.

cs.LG↗

arXiv2026

Machine Unlearning under Retain-Forget Entanglement

Forgetting a subset in machine unlearning is rarely an isolated task. Often, retained samples that are closely related to the forget set can be unintentionally affected, particularly when they share correlated features from pretraining or exhibit strong semantic similarities. To address this challenge, we propose a novel two-phase optimization framework specifically designed to handle such retai-forget entanglements. In the first phase, an augmented Lagrangian method increases the loss on the forget set while preserving accuracy on less-related retained samples. The second phase applies a gradient projection step, regularized by the Wasserstein-2 distance, to mitigate performance degradation on semantically related retained samples without compromising the unlearning objective. We validate our approach through comprehensive experiments on multiple unlearning tasks, standard benchmark datasets, and diverse neural architectures, demonstrating that it achieves effective and reliable unlearning while outperforming existing baselines in both accuracy retention and removal fidelity.

cs.LG↗

arXiv2026

Scalable learning of macroscopic stochastic dynamics

Macroscopic dynamical descriptions of complex physical systems are crucial for understanding and controlling material behavior. With the growing availability of data and compute, machine learning has become a promising alternative to first-principles methods to build accurate macroscopic models from microscopic trajectory simulations. However, for spatially extended systems, direct simulations of sufficiently large microscopic systems that inform macroscopic behavior is prohibitive. In this work, we propose a framework that learns the macroscopic dynamics of large stochastic microscopic systems using only small-system simulations. Our framework employs a partial evolution scheme to generate training data pairs by evolving large-system snapshots within local patches. We subsequently identify the closure variables associated with the macroscopic observables and learn the macroscopic dynamics using a custom loss. Furthermore, we introduce a hierarchical upsampling scheme that enables efficient generation of large-system snapshots from small-system trajectory distributions. We empirically demonstrate the accuracy and robustness of our framework through a variety of stochastic spatially extended systems, including those described by stochastic partial differential equations, idealised lattice spin systems, and a more realistic NbMoTa alloy system.

physics.comp-ph↗

arXiv2026

Deep learning and the rate of approximation by flows

We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to approximate a diffeomorphism by flows driven by a given family $\mathcal F$ of vector fields. We show that this minimal time can be identified as a geodesic distance on a sub-Finsler manifold of diffeomorphisms, where the local geometry is characterised by a variational principle involving $\mathcal F$. This connects the learning efficiency of target relationships to their compatibility with the learning architectural choice. Further, the results suggest that the key approximation mechanism in deep learning, namely the approximation of functions by composition or dynamics, differs in a fundamental way from linear approximation theory, where linear spaces and norm-based rate estimates are replaced by manifolds and geodesic distances.

cs.LG↗

arXiv2026

Terminally constrained flow-based generative models from an optimal control perspective

We address the problem of sampling from terminally constrained distributions with pre-trained flow-based generative models through an optimal control formulation. Theoretically, we characterize the value function by a Hamilton-Jacobi-Bellman equation and derive the optimal feedback control as the minimizer of the associated Hamiltonian. We show that as the control penalty increases, the controlled process recovers the reference distribution, while as the penalty vanishes, the terminal law converges to a generalized Wasserstein projection onto the constraint manifold. Algorithmically, we introduce Terminal Optimal Control with Flow-based models (TOCFlow), a geometry-aware sampling-time guidance method for pre-trained flows. Solving the control problem in a terminal co-moving frame that tracks reference trajectories yields a closed-form scalar damping factor along the Riemannian gradient, capturing second-order curvature effects without matrix inversions. TOCFlow therefore matches the geometric consistency of Gauss-Newton updates at the computational cost of standard gradient guidance. We evaluate TOCFlow on three high-dimensional scientific tasks spanning equality, inequality, and global statistical constraints, namely Darcy flow, constrained trajectory planning, and turbulence snapshot generation with Kolmogorov spectral scaling. Across all settings, TOCFlow improves constraint satisfaction over Euclidean guidance and projection baselines while preserving the reference model's generative quality.

cs.LG↗

arXiv2025

AI4X Roadmap: Artificial Intelligence for the advancement of scientific pursuit and its future directions

Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.

physics.soc-ph↗

arXiv2025

Identifiable learning of dissipative dynamics

Complex dissipative systems appear across science and engineering, from polymers and active matter to learning algorithms. These systems operate far from equilibrium, where energy dissipation and time irreversibility govern their behavior but are difficult to quantify from data. Here, we introduce a universal and identifiable neural framework that learns dissipative stochastic dynamics directly from trajectories while ensuring interpretability, expressiveness, and uniqueness. Our method identifies a unique energy landscape, separates reversible from irreversible motion, and allows direct computation of the entropy production, providing a principled measure of irreversibility and deviations from equilibrium. Applications to polymer stretching in elongational flow and to stochastic gradient Langevin dynamics reveal new insights, including super-linear scaling of barrier heights and sub-linear scaling of entropy production rates with the strain rate, and the suppression of irreversibility with increasing batch size. Our methodology thus establishes a general, data-driven framework for discovering and interpreting non-equilibrium dynamics.

cs.LG↗

arXiv2025

A unified framework for establishing the universal approximation of transformer-type architectures

We investigate the universal approximation property (UAP) of transformer-type architectures, providing a unified theoretical framework that extends prior results on residual networks to models incorporating attention mechanisms. Our work identifies token distinguishability as a fundamental requirement for UAP and introduces a general sufficient condition that applies to a broad class of architectures. Leveraging an analyticity assumption on the attention layer, we can significantly simplify the verification of this condition, providing a non-constructive approach in establishing UAP for such architectures. We demonstrate the applicability of our framework by proving UAP for transformers with various attention mechanisms, including kernel-based and sparse attention mechanisms. The corollaries of our results either generalize prior works or establish UAP for architectures not previously covered. Furthermore, our framework offers a principled foundation for designing novel transformer architectures with inherent UAP guarantees, including those with specific functional symmetries. We propose examples to illustrate these insights.

cs.LG↗