arXiv subjects

Christopher Earls

Publications and source records attributed to Christopher Earls.

14 recordsLinked to original sources

arXiv2026

Mechanistic Interpretability and Causal Feature Steering of Neural Quantum States via Sparse Autoencoders

Neural Quantum States (NQS) are a remarkably expressive class of variational ans\"atze for quantum many-body wavefunctions, yet little is understood about their internal mechanisms: trained on variational objectives alone, how do NQS accurately capture physical observables that they have never been explicitly optimized for? In this work, we present a systematic approach to analyze the internal activations of NQS using sparse autoencoders. We extract features from the residual stream and demonstrate that these features strongly correlate with physical observables such as order parameters, staggered magnetization, and half-chain correlators, across both ground state representation and real-time dynamics. Remarkably, the discovery of these features is entirely unsupervised, with no physical labels provided. We further establish that such features causally affect the corresponding observables predicted by NQS, by showing that targeted, post-training intervention on a \textit{single} feature smoothly and monotonically steers the corresponding observable, while leaving the variational energy nearly unchanged. These results demonstrate that NQS are not merely functional approximators, but encode rich, interpretable internal representations of physical information. Our approach provides both a diagnostic and an intervention tool for NQS, and serves as a foundation for using mechanistic interpretability towards more reliable, transparent NQS.

quant-ph↗

arXiv2026

Universal Neural Propagator: Learning Time Evolution in Many-Body Quantum Systems

Conventional approaches to simulating quantum many-body dynamics produce a single trajectory: if the Hamiltonian or the initial state is changed, the computation must be re-performed. Recent efforts toward foundation models have begun to address this limitation, yet existing methods transfer across either Hamiltonians or initial states, but not both. In this work, we introduce the Universal Neural Propagator (UNP), a single, unified model that learns the functional mapping from driving protocols to time-evolution propagators. Trained in an entirely self-supervised way, a single UNP model predicts dynamics across a function space of driving protocols and an exponentially large Hilbert space of initial states simultaneously. We benchmark on a two-dimensional driven Ising model and demonstrate the UNP's accuracy and transferability across product and entangled initial states, as well as for both in- and out-of-distribution driving protocols. The UNP remains accurate at system sizes beyond exact diagonalization, and can be efficiently fine-tuned across all initial states using observable data. By shifting the object of learning from quantum states to operators, this work opens a route toward transferable simulation of driven quantum matter.

quant-ph↗

arXiv2026

Neural Operator Quantum State: A Foundation Model for Quantum Dynamics

Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schr\"odinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.

quant-ph↗

arXiv2026

Attention in Krylov Space: Transformer-Based Extrapolation of Lanczos Coefficients

The Universal Operator Growth Hypothesis formulates time evolution of operators through Lanczos coefficients. In practice, however, numerical instability and memory cost limit the number of coefficients that can be exactly computed. In response to these challenges, the standard approach relies on fitting early coefficients to asymptotic forms, but such procedures can miss subleading, history-dependent structures in the coefficients that subsequently affect reconstructed observables. In this work, we treat the Lanczos coefficients as a causal time sequence and introduce a transformer-based model to autoregressively predict future Lanczos coefficients from short prefixes. For classical and quantum chaotic systems, our model outperforms asymptotic fits in both coefficient extrapolation and physical observable reconstruction, and achieves an order-of-magnitude reduction in error. The model also accurately extrapolates coefficients in integrable regimes, where no universal asymptotic fit exists. Remarkably, our model transfers across system sizes: it can be trained on smaller systems and then be used to extrapolate coefficients on a larger system \emph{without retraining}. By probing the learned attention patterns and performing targeted attention ablations, we identify portions of the coefficient history that are most influential for accurate forecasts. Our results demonstrate that modern sequence models can serve as practical surrogates for probing operator dynamics deep in Krylov space, where brute-force Lanczos iteration can be computationally prohibitive.

quant-ph↗

arXiv2025

Fourier Neural Operators for Time-Periodic Quantum Systems: Learning Floquet Hamiltonians, Observable Dynamics, and Operator Growth

Time-periodic quantum systems exhibit a rich variety of far-from-equilibrium phenomena and serve as ideal platforms for quantum engineering and control. However, simulating their dynamics with conventional numerical methods remains challenging due to the exponential growth of Hilbert space dimension and rapid spreading of entanglement. In this work, we introduce Fourier neural operators (FNOs) as an efficient, accurate, and scalable framework for nonequilibrium quantum dynamics. Parameterized in Fourier space, FNO naturally captures temporal correlations and remains minimally dependent on discretization of time. We demonstrate the versatility of FNO through three complementary learning paradigms: reconstructing effective Floquet Hamiltonians, predicting expectation values of local observables, and learning quantum information spreading. For each learning task, FNO achieves remarkable accuracy, while attaining a significant speedup, compared to exact numerical methods. Moreover, FNO possesses capabilities beyond that of conventional methods, such as predicting all local observables from a subset of measurements without information about the Hamiltonian, as well as extrapolating beyond the time window provided by training data, enabling access to observables and operator-spreading dynamics that might be beyond the coherence time. By employing a spatially local basis, we argue that the computational cost of FNOs scales only polynomially with the system size. Our results establish FNO as a versatile and scalable computation framework that integrates numerical simulations and experimental data seamlessly, with direct implications for extracting meaningful physics from measurements by near-term quantum computers.

quant-ph↗

arXiv2025

What's in a prompt? Language models encode literary style in prompt embeddings

Large language models use high-dimensional latent spaces to encode and process textual information. Much work has investigated how the conceptual content of words translates into geometrical relationships between their vector representations. Fewer studies analyze how the cumulative information of an entire prompt becomes condensed into individual embeddings under the action of transformer layers. We use literary pieces to show that information about intangible, rather than factual, aspects of the prompt are contained in deep representations. We observe that short excerpts (10 - 100 tokens) from different novels separate in the latent space independently from what next-token prediction they converge towards. Ensembles from books from the same authors are much more entangled than across authors, suggesting that embeddings encode stylistic features. This geometry of style may have applications for authorship attribution and literary analysis, but most importantly reveals the sophistication of information processing and compression accomplished by language models.

cs.CL↗

arXiv2025

chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data

In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns an Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network from which we subsequently construct a bivariate representation in a Chebyshev basis. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials.

cs.LG↗

arXiv2023

Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, $U$, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.

cs.LG↗

arXiv2023

Principled interpolation of Green's functions learned from data

We present a data-driven approach to mathematically model physical systems whose governing partial differential equations are unknown, by learning their associated Green's function. The subject systems are observed by collecting input-output pairs of system responses under excitations drawn from a Gaussian process. Two methods are proposed to learn the Green's function. In the first method, we use the proper orthogonal decomposition (POD) modes of the system as a surrogate for the eigenvectors of the Green's function, and subsequently fit the eigenvalues, using data. In the second, we employ a generalization of the randomized singular value decomposition (SVD) to operators, in order to construct a low-rank approximation to the Green's function. Then, we propose a manifold interpolation scheme, for use in an offline-online setting, where offline excitation-response data, taken at specific model parameter instances, are compressed into empirical eigenmodes. These eigenmodes are subsequently used within a manifold interpolation scheme, to uncover other suitable eigenmodes at unseen model parameters. The approximation and interpolation numerical techniques are demonstrated on several examples in one and two dimensions.

math.NA↗

arXiv2023

PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, $U$, to approximate the system response function and a sparse, trainable vector, $ξ$, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of $U$ and $ξ$ using a loss function that (1) makes $U$ approximate the system response function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $ξ$ characterizes, and (3) promotes sparsity in $ξ$ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.

cs.LG↗

arXiv2022

PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are sparse and noisy. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.

cs.LG↗

arXiv2021

A new engineering theory describing oblique free surface impact by flexible plates

Consideration of slamming loads within the structural design of planning hulls is of critical importance in ensuring adequate structural performance in order to avoid potential catastrophic consequences. However, because of the intricacy in the interplay between complex fluid flows and nonlinear structural deformations that accompany the phenomenology of slamming, a general engineering theory in slamming has yet to be uncovered, and so design relies on specialized theories. In this paper, we propose one such theory for a design case that has, until now, eluded a proper description. In pursuit of this theory, we employ a specialized implicit, partitioned fluid-structural interaction (FSI) simulation approach, in order to study the underlying physical mechanisms accompanying the oblique impact of a flexible plate during water entry. In the present work, we first present validation results from flexible plate water entry experiments, to confirm the veracity of the developed FSI solver. Subsequent to validation, we carry out a series of numerical analyses, in an effort to characterize the regimes in impact force and plate out-of-plane deformations, as a function of impact velocities and plate flexural rigidity. Finally, we use our FSI solver, as a kind of "microscope", to study the mechanistic evolution of fluid flows and elastic plate deformations that occur during slamming. Based on these observations, we propose a novel, but simple engineering theory for flexible plates obliquely impacting the water free surface (e.g. high speed porpoising water craft reentry).

physics.flu-dyn↗

arXiv2019

Analysis of heterogeneous computing approaches to simulating heat transfer in heterogeneous material

The simulation of heat flow through heterogeneous material is important for the design of structural and electronic components. Classical analytical solutions to the heat equation PDE are not known for many such domains, even those having simple geometries. The finite element method can provide approximations to a weak form continuum solution, with increasing accuracy as the number of degrees of freedom in the model increases. This comes at a cost of increased memory usage and computation time; even when taking advantage of sparse matrix techniques for the finite element system matrix. We summarize recent approaches in solving problems in structural mechanics and steady state heat conduction which do not require the explicit assembly of any system matrices, and adapt them to a method for solving the time-depended flow of heat. These approaches are highly parallelizable, and can be performed on graphical processing units (GPUs). Furthermore, they lend themselves to the simulation of heterogeneous material, with a minimum of added complexity. We present the mathematical framework of assembly-free FEM approaches, through which we summarize the benefits of GPU computation. We discuss our implementation using the OpenCL computing framework, and show how it is further adapted for use on multiple GPUs. We compare the performance of single and dual GPUs implementations of our method with previous GPU computing strategies from the literature and a CPU sparse matrix approach. The utility of the novel method is demonstrated through the solution of a real-world coefficient inverse problem that requires thousands of transient heat flow simulations, each of which involves solving a 1 million degree of freedom linear system over hundreds of time steps.

math.NA↗

arXiv2019

A subspace pursuit method to infer refractivity in the marine atmospheric boundary layer

Inferring electromagnetic propagation characteristics within the marine atmospheric boundary layer (MABL) from data in real time is crucial for modern maritime navigation and communications. The propagation of electromagnetic waves is well modeled by a partial differential equation (PDE): a Helmholtz equation. A natural way to solve the MABL characterization inverse problem is to minimize what is observed and what is predicted by the PDE. However, this optimization is difficult because it has many local minima. We propose an alternative solution that relies on the properties of the PDE but does not involve solving the full forward model. Ducted environments result in an EM field which can be decomposed into a few propagating, trapped modes. These modes are a subset of the solutions to a Sturm-Liouville eigenvalue problem. We design a new objective function that measures the distance from the observations to a subspace spanned by these eigenvectors. The resulting optimization problem is much easier than the one that arises in the standard approach, and we show how to solve the associated nonlinear eigenvalue problem efficiently, leading to a real-time method.

physics.ao-ph↗