arXiv subjects

Nicolas Boullé

Publications and source records attributed to Nicolas Boullé.

At least 19 recordsLinked to original sources

arXiv2026

The Geometry of Ignorance: LLMs Know When to Temper Bayesian Priors

What does a language model predict when it has few clues? The answer lurks in its unembedding geometry: a single direction of the unembedding matrix encodes the unigram distribution of the training corpus, which serves as the Bayesian prior the model falls back on when uncertain. This structure --- which we term the \emph{direction of ignorance} --- appears in all four model families examined (\texttt{Llama}, \texttt{Qwen}, \texttt{Gemma}, and \texttt{Pythia}), ranging from 0.4B to 405B parameters. Projecting the final prediction state onto this direction yields a per-token \emph{prior loading factor} $λ$, which, empirically, declines steadily as the context becomes more informative. Formally, the same projection decomposes the prediction state into two orthogonal vectors that correspond exactly to the two factors of a tempered Bayesian update: a unigram prior raised to the exponent $λ$ and a context-driven likelihood. This geometric-probabilistic interpretation calibrates $λ$, making it meaningfully comparable across model sizes and families, with larger models generally exhibiting lower prior reliance in the high-context limit. Finally, we show that the direction of ignorance is causally active: raising or lowering $λ$ at the final prediction state steers the prediction toward or away from the unigram prior in KL divergence.

cs.LG↗

arXiv2026

Conformal Uncertainty Quantification Guarantees for Neural Operators

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-γ$ fraction of the evaluation domain, with probability at least $1-α$ over test and calibration inputs, where $α,γ\in(0,1)$. Our method reduces a normalized residual field to its spatial $(1-γ)$-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.

math.NA↗

arXiv2026

A Graph Signal Processing Perspective on Numerical Sequence Representations in LLM In-Context Learning

Pretrained large language models (LLMs) have demonstrated in-context learning (ICL) capabilities for numerical inference over sequences serialized as text. Prior work has identified and characterized this form of numerical inference primarily through output-level evaluations such as prediction error. However, how numerical information is organized within LLM representations remains much less understood. To study this internal organization, we adopt a graph signal processing perspective in which attention induces a weighted graph over tokens, while token hidden states define signals on its nodes. Quantitative graph-spectral diagnostics and qualitative token-graph visualizations reveal that representations become more clearly differentiated by input dynamical complexity as context length increases. Simpler inputs produce attention-induced token graphs with stronger global connectivity and smoother, spectrally concentrated hidden-state signals, whereas more complex inputs produce more localized graphs and hidden-state signals with broader spectral support and greater high-frequency energy. Together, these findings point to systematic, context-dependent internal signatures associated with numerical ICL that are conserved across model families.

cs.LG↗

arXiv2026

Physics-guided correction for operator learning under model misspecification

Physics-informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics-guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. We realize the operator model correction using a serial DeepONet architecture, where the first DeepONet provides a solution prediction and the second DeepONet learns an additive correction conditioned on both the input function and the solution prediction. The learned correction is incorporated into the physics residual and trained together with data-consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion-reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.

math.NA↗

arXiv2026

A zero-one law for one-shot system identification

Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differential operators and dynamical systems. For a single input-response pair, recovery is possible exactly when the evaluated dictionary terms are linearly independent. We prove a sharp zero-one law: either no input uniquely determines the coefficients, or almost every random input sampled from a nondegenerate Gaussian measure does. This dichotomy reduces one-shot system identification to a question about degenerate inputs and provides an a posteriori certificate for any recovered model. Numerical examples recover dynamical systems, nonlinear partial differential equations, and structured matrix families from single trajectory data, while also detecting when an extra probe is necessary.

math.NA↗

arXiv2026

Jacobian Scopes: token-level causal attributions in LLMs

Large language models (LLMs) make next-token predictions based on clues present in their context, such as semantic descriptions and in-context examples. Yet, elucidating which prior tokens most strongly influence a given prediction remains challenging due to the proliferation of layers and attention heads in modern architectures. We propose Jacobian Scopes, a suite of gradient-based, token-level causal attribution methods for interpreting LLM predictions. Grounded in perturbation theory and information geometry, Jacobian Scopes quantify how input tokens influence various aspects of a model's prediction, such as specific logits, the full predictive distribution, and model uncertainty (effective temperature). Through case studies spanning instruction understanding, translation, and in-context learning (ICL), we demonstrate how Jacobian Scopes reveal implicit political biases, uncover word- and phrase-level translation strategies, and shed light on recently debated mechanisms underlying in-context time-series forecasting. To facilitate exploration of Jacobian Scopes on custom text, we open-source our implementations and provide a cloud-hosted interactive demo at https://huggingface.co/spaces/Typony/JacobianScopes.

cs.CL↗

arXiv2026

Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

cs.LG↗

arXiv2026

Generalized Discrete Diffusion from Snapshots

We introduce Generalized Discrete Diffusion from Snapshots (GDDS), a unified framework for discrete diffusion modeling that supports arbitrary noising processes over large discrete state spaces. Our formulation encompasses all existing discrete diffusion approaches, while allowing significantly greater flexibility in the choice of corruption dynamics. The forward noising process relies on uniformization and enables fast arbitrary corruption. For the reverse process, we derive a simple evidence lower bound (ELBO) based on snapshot latents, instead of the entire noising path, that allows efficient training of standard generative modeling architectures with clear probabilistic interpretation. Our experiments on large-vocabulary discrete generation tasks suggest that the proposed framework outperforms existing discrete diffusion methods in terms of training efficiency and generation quality, and beats autoregressive models for the first time at this scale. We provide the code along with a blog post on the project page : \href{https://oussamazekri.fr/gdds}{https://oussamazekri.fr/gdds}.

stat.ML↗

arXiv2026

Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.

math.NA↗

arXiv2026

Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism

Large language models (LLMs) have demonstrated emergent in-context learning (ICL) capabilities across a range of tasks, including zero-shot time-series forecasting. We show that text-trained foundation models can accurately extrapolate spatiotemporal dynamics from discretized partial differential equation (PDE) solutions without fine-tuning or natural language prompting. Predictive accuracy improves with longer temporal contexts but degrades at finer spatial discretizations. In multi-step rollouts, where the model recursively predicts future spatial states over multiple time steps, errors grow algebraically with the time horizon, reminiscent of global error accumulation in classical finite-difference solvers. We interpret these trends as in-context neural scaling laws, where prediction quality varies predictably with both context length and output length. To better understand how LLMs are able to internally process PDE solutions so as to accurately roll them out, we analyze token-level output distributions and uncover a consistent three-stage ICL progression: beginning with syntactic pattern imitation, transitioning through an exploratory high-entropy phase, and culminating in confident, numerically grounded predictions.

cs.LG↗

arXiv2026

Multi-Level Monte Carlo Training of Neural Operators

Operator learning is a rapidly growing field that aims to approximate nonlinear operators related to partial differential equations (PDEs) using neural operators. These rely on discretization of input and output functions and are, usually, expensive to train for large-scale problems at high-resolution. Motivated by this, we present a Multi-Level Monte Carlo (MLMC) approach to train neural operators by leveraging a hierarchy of resolutions of function discretization. Our framework relies on using gradient corrections from fewer samples of fine-resolution data to decrease the computational cost of training while maintaining a high level accuracy. The proposed MLMC training procedure can be applied to any architecture accepting multi-resolution data. Our numerical experiments on a range of state-of-the-art models and test-cases demonstrate improved computational efficiency compared to traditional single-resolution training approaches, and highlight the existence of a Pareto curve between accuracy and computational time, related to the number of samples per resolution.

cs.LG↗

arXiv2025

Fine-Tuning Discrete Diffusion Models with Policy Gradient Methods

Discrete diffusion models have recently gained significant attention due to their ability to process complex discrete structures for language modeling. However, fine-tuning these models with policy gradient methods, as is commonly done in Reinforcement Learning from Human Feedback (RLHF), remains a challenging task. We propose an efficient, broadly applicable, and theoretically justified policy gradient algorithm, called Score Entropy Policy Optimization (\SEPO), for fine-tuning discrete diffusion models over non-differentiable rewards. Our numerical experiments across several discrete generative tasks demonstrate the scalability and efficiency of our method. Our code is available at https://github.com/ozekri/SEPO.

stat.ML↗

arXiv2025

Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces

Data-driven spectral analysis of Koopman operators is a powerful tool for understanding numerous real-world dynamical systems, from neuronal activity to variations in sea surface temperature. The Koopman operator acts on a function space and is most commonly studied on the space of square-integrable functions. However, defining it on a suitable reproducing kernel Hilbert space (RKHS) offers numerous practical advantages, including pointwise predictions with error bounds, improved spectral properties that facilitate computations, and more efficient algorithms, particularly in high dimensions. We introduce the first general, provably convergent, data-driven algorithms for computing spectral properties of Koopman and Perron--Frobenius operators on RKHSs. These methods efficiently compute spectra and pseudospectra with error control and spectral measures while exploiting the RKHS structure to avoid the large-data limits required in the $L^2$ settings. The function space is determined by a user-specified kernel, eliminating the need for quadrature-based sampling as in $L^2$ and enabling greater flexibility with finite, externally provided datasets. Using the Solvability Complexity Index hierarchy, we construct adversarial dynamical systems for these problems to show that no algorithm can succeed in fewer limits, thereby proving the optimality of our algorithms. Notably, this impossibility extends to randomized algorithms and datasets. We demonstrate the effectiveness of our algorithms on challenging, high-dimensional datasets arising from real-world measurements and high-fidelity numerical simulations, including turbulent channel flow, molecular dynamics of a binding protein, Antarctic sea ice concentration, and Northern Hemisphere sea surface height. The algorithms are publicly available in the software package $\texttt{SpecRKHS}$.

math.NA↗

arXiv2025

Multiplicative Dynamic Mode Decomposition

Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities. Recent methods have aimed to approximate Koopman operators while preserving key structures. However, approximating Koopman operators typically requires a dictionary of observables to capture the system's behavior in a finite-dimensional subspace. The selection of these functions is often heuristic, may result in the loss of spectral information, and can severely complicate structure preservation. This paper introduces Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. Leveraging this multiplicative property, we guide the selection of observables and define a constrained optimization problem for the matrix approximation, which can be efficiently solved. MultDMD presents a structured approach to finite-dimensional approximations and can more accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties. The efficacy of MultDMD is demonstrated through several examples, including the nonlinear pendulum, the Lorenz system, and fluid dynamics data, where we demonstrate its remarkable robustness to noise.

math.DS↗

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

Density estimation with LLMs: a geometric investigation of in-context learning trajectories

Large language models (LLMs) demonstrate remarkable emergent abilities to perform in-context learning across various tasks, including time series forecasting. This work investigates LLMs' ability to estimate probability density functions (PDFs) from data observed in-context; such density estimation (DE) is a fundamental task underlying many probabilistic modeling problems. We leverage the Intensive Principal Component Analysis (InPCA) to visualize and analyze the in-context learning dynamics of LLaMA-2 models. Our main finding is that these LLMs all follow similar learning trajectories in a low-dimensional InPCA space, which are distinct from those of traditional density estimation methods like histograms and Gaussian kernel density estimation (KDE). We interpret the LLaMA in-context DE process as a KDE with an adaptive kernel width and shape. This custom kernel model captures a significant portion of LLaMA's behavior despite having only two parameters. We further speculate on why LLaMA's kernel width and shape differs from classical algorithms, providing insights into the mechanism of in-context probabilistic reasoning in LLMs. Our codebase, along with a 3D visualization of an LLM's in-context learning trajectory, is publicly available at https://github.com/AntonioLiu97/LLMICL_inPCA

cs.LG↗

arXiv2025

Lines of Thought in Large Language Models

Large Language Models achieve next-token prediction by transporting a vectorized piece of text (prompt) across an accompanying embedding space under the action of successive transformer layers. The resulting high-dimensional trajectories realize different contextualization, or 'thinking', steps, and fully determine the output probability distribution. We aim to characterize the statistical properties of ensembles of these 'lines of thought.' We observe that independent trajectories cluster along a low-dimensional, non-Euclidean manifold, and that their path can be well approximated by a stochastic equation with few parameters extracted from data. We find it remarkable that the vast complexity of such large models can be reduced to a much simpler form, and we reflect on implications.

cs.LG↗

arXiv2025

Large Language Models as Markov Chains

Large language models (LLMs) are remarkably efficient across a wide range of natural language processing tasks and well beyond them. However, a comprehensive theoretical analysis of the LLMs' generalization capabilities remains elusive. In our paper, we approach this task by drawing an equivalence between autoregressive transformer-based language models and Markov chains defined on a finite state space. This allows us to study the multi-step inference mechanism of LLMs from first principles. We relate the obtained results to the pathological behavior observed with LLMs such as repetitions and incoherent replies with high temperature. Finally, we leverage the proposed formalization to derive pre-training and in-context learning generalization bounds for LLMs under realistic data and model assumptions. Experiments with the most recent Llama and Gemma herds of models show that our theory correctly captures their behavior in practice.

stat.ML↗