arXiv subjects

Ioannis G. Kevrekidis

Publications and source records attributed to Ioannis G. Kevrekidis.

At least 19 recordsLinked to original sources

arXiv2026

Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

We present a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solutions are sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are shallow, single-hidden-layer networks with randomly sampled and fixed a priori hidden-layer weights; only the linear output layer weights are optimized, reducing training to a single least-squares solve. This linear output structure enables the explicit generalized eigenvalue formulation governing the linear stability of stationary solutions, separating the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions (BCs), at no additional training cost and without requiring additional PDE solves. However, the random projection collocation matrix is inherently numerically rank-deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near-zero modes. To overcome this limitation, we introduce a matrix-free shift-invert Krylov-Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank-deficient collocation matrix and enabling the reliable computation of several leading eigenpairs of the physical Jacobian - the discretized Frechet derivative of the PDE operator with respect to the solution field, whose eigenvalue spectrum determines linear stability. We further prove that the PI-RPNN-based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions, rigorously explaining the observed ill-conditioning.

math.NA↗

arXiv2026

A Theoretical Analysis of Generalization Dynamics in Neural Networks under Gradient Descent with Weight Decay

Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the $\ell^2$ loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.

cs.LG↗

arXiv2026

Generative Learning of Separatrices

The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.

cs.LG↗

arXiv2026

A Mechanistic Analysis of Transformers for Dynamical Systems

Transformers are increasingly adopted for modeling and forecasting time-series, yet their internal mechanisms remain poorly understood from a dynamical systems perspective. In contrast to classical autoregressive and state-space models, which benefit from well-established theoretical foundations, Transformer architectures are typically treated as black boxes. This gap becomes particularly relevant as attention-based models are considered for general-purpose or zero-shot forecasting across diverse dynamical regimes. In this work, we do not propose a new forecasting model, but instead investigate the representational capabilities and limitations of single-layer Transformers when applied to dynamical data. Building on a dynamical systems perspective, we interpret causal self-attention as a linear, history-dependent recurrence and analyze how it processes temporal information. Through a series of linear and nonlinear case studies, we identify distinct operational regimes. For linear systems, we show that in the single-head attention-only setting, the convexity constraint imposed by softmax attention restricts the class of autoregressive operators that can be represented, leading to oversmoothing when the target dynamics require mixed-sign lag coefficients. For nonlinear systems under partial observability, attention instead acts as an adaptive delay-embedding mechanism, enabling effective state reconstruction when sufficient temporal context and latent dimensionality are available. These results help bridge empirical observations with classical dynamical systems theory, providing insight into when and why Transformers succeed or fail as models of dynamical systems.

cs.LG↗

arXiv2026

Blow-up Parameter Landscapes for Polynomial Dynamical Systems

Finite-time blow-up is one of the ways in which a dynamical model can become singular, often signaling the breakdown of either the modeled physical system or the model itself. Determining whether blow-up occurs, and for which parameter values and initial conditions, is therefore a fundamental problem in the analysis of nonlinear dynamical systems. We develop a numerical framework for identifying regions of parameter space in which a dynamical system governed by a system of first-order ordinary differential equations with polynomial right-hand sides exhibits finite-time blow-up for at least one initial condition. The approach combines compactification of the phase space with computational algebraic techniques, producing partitioned parameter landscapes that reveal blow-up and non-blow-up regimes. Through several examples, we show that the method replaces problem-specific hand calculations with an automated computational tool for analyzing blow-up regions in parameter-dependent dynamical systems.

math.DS↗

arXiv2026

Singularities in Multi-Objective Optimization and their Crossing during Continuation

Continuation methods help trace Pareto sets in multi-objective optimization but are inherently local: a single run traces a single connected branch, requiring multiple restarts to recover disconnected components of Pareto fronts. We show that, for unconstrained bi-objective problems under weighted-sum scalarization, these disconnects can be artifacts of singularities in the scalarization parameter, where the weight $λ$ diverges as the objective gradients become collinear. Recasting Pareto optimality as a nonlinear system, we apply pseudo-arclength continuation to follow the Pareto-critical set, and show that suitable singular reparameterizations allow crossing these singularities in systematically, recovering disconnected branches in a single run. A coordinate-wise projective compactification further provides a unified framework for parameter and decision-space variables. We demonstrate the approach on the ZDT3 benchmark and modifications.

math.OC↗

arXiv2026

FlowEdit: Information-Theoretic Control of LLM Reasoning Flows for Ill-posed Problems Involving Conflicts

Large Language Models (LLMs) perform strongly on well-specified reasoning tasks with a feasible answer. However, problems encountered in the open world can become ill-posed due to inconsistent conditions, conflicting statements, or mutually incompatible requirements, admitting no valid responses. We argue that reasoning of such ill-posed problems involving conflicts require novel LLM capabilities to make hidden conflicts explicit, maintain competing hypotheses via multiple reasoning branches, and generate alternative responses in a single pass, all of which are challenging due to the limitation of the next-token prediction mechanism in LLMs. To this end, we propose FlowEdit, a novel framework that leverages information-theoretic principles to quantify and regulate internal reasoning flows of LLMs, for generating a full set of alternative responses under valid hypotheses. FlowEdit can be viewed as enforcing a branch-aware reasoning process using two dual information-theoretic objectives on the model's internal reasoning representations: maximizing the information flow from each selected hypothesis to the branch outcome, while minimizing the overlap and conditional dependence across sibling branches, to provide a diverse, informative set of responses with broad coverage. We show that this is achieved through tractable variational bounds under boundary embeddings being ε-sufficient, optimizing the underlying conditional mutual information in LLM reasoning process. Extensive experiments demonstrate that FlowEdit outperforms leading proprietary models, improving exact-set-match accuracy by 68%, while boosting overall response informativeness by 24%. We further show that flow regulation surfaces in the token stream as a redistribution of next-token entropy that concentrates inside each branch, amplifies at flow boundaries, and scales with the number of flows the problem requires.

cs.AI↗

arXiv2026

Learning to Choose: An Empowerment-Guided Multi-Agent System with semantic communication for Adaptive Method Selection

Automating scientific computing workflows requires more than generating executable code: autonomous systems must also select appropriate computational strategies, implement them faithfully, and ensure that the resulting outcomes remain causally attributable to the decisions that produced them. In multi-agent pipelines, this process is particularly fragile, as small inconsistencies between agent intentions and actions can lead to semantic drift, where the eventually executed procedure no longer reflects the originally selected strategy, thereby corrupting downstream evaluation and adaptation. In this work, motivated by the ATHENA framework (Toscano et al., 2025; Toscano et al., 2026) and the concept of empowerment (Yiu et al., 2025), we introduce a multi-agent framework that combines contextual bandits with structured inter-agent communication and, most importantly, semantic checkpoints that preserve action-outcome fidelity throughout the pipeline. The system integrates specialized large language model (LLM) agents, grounded code generation, and self-healing execution loops within an adaptive decision-making architecture. Interpreting the framework through the lens of empowerment, we show that reliable autonomous learning requires not only identifying high-quality actions, but also preserving the integrity of their propagation across agents. Using sensitivity analysis and uncertainty quantification workflows as representative case studies, we demonstrate that unchecked semantic drift degrades policy learning, whereas the proposed framework improves convergence, robustness, and adaptation to novel problem contexts. These results suggest a broader design principle for scientific multi-agent systems: adaptive decision-making must be coupled with explicit mechanisms that guarantee semantic consistency and reliable information flow across the computational pipeline.

cs.AI↗

arXiv2026

Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators

The control of high-dimensional distributed parameter systems (DPS) remains a challenge when explicit coarse-grained equations are unavailable. Classical equation-free (EF) approaches rely on fine-scale simulators treated as black-box timesteppers. However, repeated simulations for steady-state computation, linearization, and control design are often computationally prohibitive, or the microscopic timestepper may not even be available, leaving us with data as the only resource. We propose a data-driven alternative that uses local neural operators, trained on spatiotemporal microscopic/mesoscopic data, to obtain efficient short-time solution operators. These surrogates are employed within Krylov subspace methods to compute coarse stable and unstable steady states, while also providing Jacobian information in a matrix-free manner. Krylov-Arnoldi iterations then approximate the dominant eigenspectrum, yielding reduced models that capture the open-loop slow dynamics without explicit Jacobian assembly. Both discrete-time Linear Quadratic Regulator (dLQR) and pole-placement (PP) controllers are based on this reduced system and lifted back to the full nonlinear dynamics, thereby closing the feedback loop. The framework is validated by stabilizing an unstable steady-state of the Liouville-Bratu PDE, demonstrating consistent performance between the learned surrogate and the true system, with quantified degradation under plant-model mismatch.

eess.SY↗

arXiv2026

Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks

We propose a hybrid physics-informed machine learning framework to approximate invariant manifolds (IMs) of discrete-time dynamical systems driven by exogenous autonomous dynamics (exosystems). Such systems appear in applications ranging from control theory to modeling collective multi-agent behavior (e.g., bird flocks, traffic dynamics) under hierarchical leadership. The IM learning problem is formulated as solving nonlinear functional equations derived from the invariance equation, expressing the manifold as a relationship between exogenous and system states. The proposed approach combines polynomial series with shallow neural networks, leveraging their complementary strengths. We focus on low- to medium-dimensional manifolds where polynomial expansions remain tractable. Near equilibrium, polynomial series provide interpretability and convergence, while farther away neural networks capture global structure through their universal approximation capability. A continuity penalty enforces consistency between both representations at their interface, and training is performed using analytically derived derivatives within the Levenberg-Marquardt scheme. Naturally, depending on the dimensionality of the input-driven system, one may also employ a purely neural network-based IM approximation, for which we also establish a universal approximation theorem based on certain assumptions on system dynamics. The framework is evaluated on two benchmark problems: an enzymatic bioreactor and a leader-follower car-following model. We analyze convergence, approximation accuracy, and computational cost, and compare standalone neural networks, polynomial expansions, and the hybrid method. Results show that the hybrid approach achieves superior accuracy compared to standalone schemes.

math.NA↗

arXiv2026

Dictionary learning for Kernel EDMD

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD we show how to perform gradient-based optimization over the learnable kernel parameters, and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky PDE, showcasing the method's different strengths.

math.DS↗

arXiv2026

BumpNet: A Sparse MLP Framework for Learning PDE Solutions

We introduce BumpNet, a sparse multilayer perceptron (MLP) framework for PDE numerical solution and operator learning. BumpNet is based on basis function expansion, which makes them superficially similar to radial-basis function (RBF) networks. However, the basis functions in BumpNet are constructed from ordinary sigmoid activation functions in a sparse multi-layer framework. This makes BumpNet a MLP, not a RBF neural network, enabling the efficient use of modern training techniques optimized for MLPs. All parameters of the basis functions, including shape, location, and amplitude, are fully trainable. Model parsimony is encouraged through a basis function pruning scheme. BumpNet is a general meshless framework that can be combined with existing neural architectures for learning PDE solutions: here, we propose Bump-PINNs (BumpNet with physics-informed neural networks) for solving general PDEs; Bump-EDNN (BumpNet with evolutionary deep neural networks) to solve time-evolution PDEs; and Bump-DeepONet (BumpNet with deep operator networks) for PDE operator learning. We prove that BumpNets and Bump-DeepONets are universal approximators of continuous functions and continuous operators, respectively. Bump-PINNs are trained using the same collocation-based approach used by PINNs; Bump-EDNN uses a BumpNet only in the spatial domain and uses EDNNs to advance the solution in time; while Bump-DeepONets employ a BumpNet regression network as the trunk network of a DeepONet. Extensive numerical experiments demonstrate the efficiency and accuracy of BumpNets.

cs.LG↗

arXiv2026

Optimal Transport, Timesteppers, Newton-Krylov Methods and Steady States of Collective Particle Dynamics

Timesteppers constitute a powerful tool in modern computational science and engineering. Although they are typically used to advance the system forward in time, they can also be viewed as nonlinear mappings that implicitly encode steady states and stability information. In this work, we present an extension of the matrix-free framework for calculating, via timesteppers, steady states of deterministic systems to stochastic particle simulations, where intrinsic randomness prevents direct steady state extraction. By formulating stochastic timesteppers in the language of optimal transport, we reinterpret them as operators acting on probability measures rather than on individual particle trajectories. This perspective enables the construction of smooth cumulative- and inverse-cumulative-distribution-function ((I)CDF) timesteppers that evolve distributions rather than particles. Combined with matrix-free Newton-Krylov solvers, these smooth timesteppers allow efficient computation of steady-state distributions even under high stochastic noise. We perform an error analysis quantifying how noise affects finite-difference Jacobian action approximations, and demonstrate that convergence can be obtained even in high noise regimes. Finally, we introduce higher-dimensional generalizations based on smooth CDF-related representations of particles and validate their performance on a non-trivial two-dimensional distribution. Together, these developments establish a unified variational framework for computing meaningful steady states of both deterministic and stochastic timesteppers.

math.NA↗

arXiv2025

Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems

We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics.

math.NA↗

arXiv2025

A Physics-informed Multi-resolution Neural Operator

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

cs.LG↗

arXiv2025

Enabling Local Neural Operators to perform Equation-Free System-Level Analysis

Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that bypass both the explicit equation identification and their subsequent numerical solving. Still, NOs have so far primarily been employed to explore the dynamical behavior as surrogates of brute-force temporal simulations/predictions. Their potential for systematic rigorous numerical system-level tasks, such as fixed-point, stability, and bifurcation analysis - crucial for predicting irreversible transitions in real-world phenomena - remains largely unexplored. Toward this aim, inspired by the Equation-Free multiscale framework, we propose and implement a framework that integrates (local) NOs with advanced iterative numerical methods in the Krylov subspace, so as to perform efficient system-level stability and bifurcation analysis of large-scale dynamical systems. Beyond fixed point, stability, and bifurcation analysis enabled by local in time NOs, we also demonstrate the usefulness of local in space as well as in space-time ("patch") NOs in accelerating the computer-aided analysis of spatiotemporal dynamics. We illustrate our framework via three nonlinear PDE benchmarks: the 1D Allen-Cahn equation, which undergoes multiple concatenated pitchfork bifurcations; the Liouville-Bratu-Gelfand PDE, which features a saddle-node tipping point; and the FitzHugh-Nagumo (FHN) model, consisting of two coupled PDEs that exhibit both Hopf and saddle-node bifurcations.

cs.LG↗

arXiv2025

Neural network-based singularity detection and applications

We present a method for constructing a special type of shallow neural network that learns univariate meromorphic functions with pole-type singularities. Our method is based on using a finite set of Laurent coefficients as input information, which we compute by FFT, employing values of the investigated function on some contour $Γ$ in the complex plane. The primary components of our methodology are the following: (1) the adaptive construction of rational polynomial activation functions, (2) a novel backpropagation-free method for determining the weights and biases of the hidden layer, and (3) the computation of the weights and biases of the output layer through least-squares fitting. Breaking with the idea of "safe" rational activation functions, we introduce a rational activation function as a meromorphic function with a single pole situated within the domain of investigation. Employing the weights and biases of the hidden layer, we then scale and shift the pole of the activation function to find the estimated locations of the singularities; this implies that the number of neurons in the hidden layer is determined by the number of singularities of the function that is being approximated. While the weights and biases of the hidden layer are tuned so as to capture the singularities, the least-squares fitting for the computation of weights and biases of the output layer ensures approximation of the function in the rest of the domain. Through the use of Laurent-Padé rational approximation concepts, we prove locally uniform convergence of our method. We illustrate the effectiveness of our method through numerical experiments, including the construction of extensions of the time-dependent solutions of nonlinear autonomous PDEs into the complex plane, and study the dynamics of their singularities.

math.NA↗

arXiv2025

On Some Tunable Multi-fidelity Bayesian Optimization Frameworks

Multi-fidelity optimization employs surrogate models that integrate information from varying levels of fidelity to guide efficient exploration of complex design spaces while minimizing the reliance on (expensive) high-fidelity objective function evaluations. To advance Gaussian Process (GP)-based multi-fidelity optimization, we implement a proximity-based acquisition strategy that simplifies fidelity selection by eliminating the need for separate acquisition functions at each fidelity level. We also enable multi-fidelity Upper Confidence Bound (UCB) strategies by combining them with multi-fidelity GPs rather than the standard GPs typically used. We benchmark these approaches alongside other multi-fidelity acquisition strategies (including fidelity-weighted approaches) comparing their performance, reliance on high-fidelity evaluations, and hyperparameter tunability in representative optimization tasks. The results highlight the capability of the proximity-based multi-fidelity acquisition function to deliver consistent control over high-fidelity usage while maintaining convergence efficiency. Our illustrative examples include multi-fidelity chemical kinetic models, both homogeneous and heterogeneous (dynamic catalysis for ammonia production).

cs.LG↗