arXiv subjects

Lucas Lacasa

Publications and source records attributed to Lucas Lacasa.

At least 19 recordsLinked to original sources

arXiv2026

Visibility graph-based characterization of extreme values in time series

Complex dynamical systems often display extreme fluctuations of an observed variable that constitute significant deviations from the long-term average, and which are often associated with severe impacts on the system. By definition, extreme events are therefore usually explored from time series recordings. In this work, we characterize extreme values in time series using visibility graphs, a method that non-parametrically maps a time series onto a network, whose topological structure is known to inherit important characteristics of the original time series dynamics. Unlike threshold-based approaches, extreme values in this framework can be identified without the need to introduce external parameters and can be applied to time series generated by both stationary and nonstationary processes. For stationary processes, we exploit a known property of visibility graphs in which the degree of a node is monotonically and nonlinearly related to the corresponding data value. This nonlinear amplification enhances the contribution of large values while suppressing noise, while the monotonic relationship enables a direct ranking of data points according to node degree. This procedure identifies global extreme values and locally prominent ones. For nonstationary processes, the degree ranking in the visibility graph still provides a robust indicator of relative importance. We validate our findings with synthetic time series and with real climatological data. Our results show that extreme-value characterization in stationary time series is enhanced when combining standard methods with visibility-graph-based detection, whereas for nonstationary data, where conventional approaches are often ill-posed, visibility graphs provide an effective alternative. We discuss how sub-sampling the time series using only peak values preserves the ability to identify extreme values while reducing computational cost.

physics.data-an↗

arXiv2026

Scalar Representations of Neural Network Training Dynamics

Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging. In this work, we treat such training trajectories as temporal networks and apply recently proposed strategies for the scalar embedding of temporal networks. We investigate whether such a scalar embedding provides a meaningful low-dimensional representation of neural network training dynamics. Using a multilayer perceptron trained on the MNIST classification task, we show that the embedding preserves the main dynamical features observed in the original parameter space, including the emergence of sensitivity to initial conditions for specific learning rate regimes and an accurate reconstruction of the network's maximum Lyapunov exponent. We then use the embedded scalar trajectory to define a characteristic time, analogous to a Lyapunov time, after which the exponential separation between initially close embedded trajectories saturates. This characteristic time captures the typical decorrelation time between initially close network trajectories in the original high-dimensional system. Finally, we investigate the statistical organization of asymptotic training states through a spacing observable defined in the embedded space. We find that the distributions of rescaled asymptotic spacings collapse onto a common form across initial conditions and are compatible with a skew lognormal distribution. Altogether, our results suggest that scalar low-dimensional embeddings provide a useful framework for studying and visualizing the dynamical properties of neural network optimization trajectories.

cs.LG↗

arXiv2026

Kolmogorov Arnold networks (KAN) for aerodynamic prediction: a comparison with MLPs and GNNs

Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (MLPs). This architecture builds on the Kolmogorov-Arnold theorem, which endows it with universal approximation properties. While the advent of KANs has been received with excitement, there is a current debate about the possible KAN supremacy over deep multilayer perceptrons (MLPs) for classic fields such as symbolic regression, generic-purpose machine learning, natural language processing or computer vision. Here we assess the performance of KANs --and its nuanced comparison against MLPs and graph neural networks (GNNs)-- in the realm of fluid dynamics surrogate modelling. To that aim, we consider the task of predicting the surface pressure distribution over subsonic and transonic airfoils, a canonical task in aerodynamics. Our results show that KAN models show good performance in predicting the whole pressure coefficients and is able to interpolate across Mach numbers and angles of attack, however its performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense or requiring lengthier training. While the optimal KAN model have typically much lower complexity than MLP and GNN --hence resulting in faster training--, we find that KANs suffer from training instabilities, and their performance is highly dependent on a proper hyperparameter optimisation.

cs.LG↗

arXiv2026

FluidFlow: a flow-matching generative model for fluid dynamics surrogates on unstructured meshes

Computational fluid dynamics (CFD) provides high-fidelity simulations of fluid flows but remains computationally expensive for many-query applications. In recent years deep learning (DL) has been used to construct data-driven fluid-dynamic surrogate models. In this work we consider a different learning paradigm and embrace generative modelling as a framework for constructing scalable fluid-dynamics surrogate models. We introduce FluidFlow, a generative model based on conditional flow-matching, a recent alternative to diffusion models that learns deterministic transport maps between noise and data distributions. FluidFlow is specifically designed to operate directly on CFD data defined on both structured and unstructured meshes alike, without the needs to perform any mesh interpolation pre-processing and preserving geometric fidelity. We assess the capabilities of FluidFlow using two different core neural network architectures, a U-Net and diffusion transformer (DiT), and condition their learning on physically meaningful parameters. The methodology is validated on two benchmark problems of increasing complexity: prediction of pressure coefficients along an airfoil boundary across different operating conditions, and prediction of pressure and friction coefficients over a full three-dimensional aircraft geometry discretized on a large unstructured mesh. In both cases, FluidFlow outperform strong multilayer perceptron baselines, achieving significantly lower error metrics and improved generalisation across operating conditions. Notably, the transformer-based architecture enables scalable learning on large unstructured datasets while maintaining high predictive accuracy. These results demonstrate that flow-matching generative models provide an effective and flexible framework for surrogate modelling in fluid dynamics, with potential for realistic engineering and scientific applications.

cs.LG↗

arXiv2026

Leveraging chaotic transients in the training of artificial neural networks

Traditional algorithms to optimize artificial neural networks when confronted with a supervised learning task are usually exploitation-type relaxational dynamics such as gradient descent (GD). Here, we explore the dynamics of the neural network trajectory along training for unconventionally large learning rates. We show that for a region of values of the learning rate, the GD optimization shifts away from purely exploitation-like algorithm into a regime of exploration-exploitation balance, as the neural network is still capable of learning but the trajectory shows sensitive dependence on initial conditions --as characterized by positive network maximum Lyapunov exponent--. Interestingly, the characteristic training time required to reach an acceptable accuracy in the test set reaches a minimum precisely in such learning rate region, further suggesting that one can accelerate the training of artificial neural networks by locating at the onset of chaos. Our results --initially illustrated for the MNIST classification task-- qualitatively hold for a range of supervised learning tasks, {learning architectures (including both shallow and deep multilayer perceptrons and convolutional neural networks) and other hyperparameters (different activation functions and weight regularisation),} and showcase the emergent, constructive role of transient chaotic dynamics in the training of artificial neural networks.

cs.LG↗

arXiv2026

On the equivalence between nonlinear graph-based dynamics and linear dynamics on higher-order networks

In network science, collective dynamics of complex systems are typically modelled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (i.e. hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this work we discuss such structural--dynamical representation duality, and investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. Using Carleman linearisation arguments, we show that finite polynomial dynamics defined in the $|V|$ vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order $|V|$, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.

physics.soc-ph↗

arXiv2026

Fluid dynamics meet network science: two cases of temporal network eigendecomposition

Temporal networks, defined as sequences of time-aggregated adjacency matrices, sample latent graph dynamics and trace trajectories in graph space. By interpreting each adjacency matrix as a different time snapshot of a scalar field, fluid-mechanics theories can be applied to construct two distinct eigendecompositions of temporal networks. The first builds on the proper orthogonal decomposition (POD) of flowfields and decomposes the evolution of a network in terms of a basis of orthogonal network eigenmodes which are ordered in terms of their relative importance, hence enabling compression of temporal networks as well as their reconstruction from low-dimensional embeddings. The second proposes a numerical approximation of the Koopman operator, a linear operator acting on a suitable observable of the graph space which provides the best linear approximation of the latent graph dynamics. Its eigendecomposition provides a data-driven spectral description of the temporal network dynamics, in terms of dynamic modes which grow, decay or oscillate over time. Both eigendecompositions are illustrated and validated in a suite of synthetic generative models of temporal networks with varying complexity.

physics.data-an↗

arXiv2026

On the Role of Consistency Between Physics and Data in Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have gained significant attention as a surrogate modeling strategy for partial differential equations (PDEs), particularly in regimes where labeled data are scarce and physical constraints can be leveraged to regularize the learning process. In practice, however, PINNs are frequently trained using experimental or numerical data that are not fully consistent with the governing equations due to measurement noise, discretization errors, or modeling assumptions. The implications of such data-to-PDE inconsistencies on the accuracy and convergence of PINNs remain insufficiently understood. In this work, we systematically analyze how data inconsistency fundamentally limits the attainable accuracy of PINNs. We introduce the concept of a consistency barrier, defined as an intrinsic lower bound on the error that arises from mismatches between the fidelity of the data and the exact enforcement of the PDE residual. To isolate and quantify this effect, we consider the 1D viscous Burgers equation with a manufactured analytical solution, which enables full control over data fidelity and residual errors. PINNs are trained using datasets of progressively increasing numerical accuracy, as well as perfectly consistent analytical data. Results show that while the inclusion of the PDE residual allows PINNs to partially mitigate low-fidelity data and recover the dominant physical structure, the training process ultimately saturates at an error level dictated by the data inconsistency. When high-fidelity numerical data are employed, PINN solutions become indistinguishable from those trained on analytical data, indicating that the consistency barrier is effectively removed. These findings clarify the interplay between data quality and physics enforcement in PINNs providing practical guidance for the construction and interpretation of physics-informed surrogate models.

cs.LG↗

arXiv2025

Reliable Statistical Guarantees for Conformal Predictors with Small Datasets

Surrogate models (including deep neural networks and other machine learning algorithms in supervised learning) are capable of approximating arbitrarily complex, high-dimensional input-output problems in science and engineering, but require a thorough data-agnostic uncertainty quantification analysis before these can be deployed for any safety-critical application. The standard approach for data-agnostic uncertainty quantification is to use conformal prediction (CP), a well-established framework to build uncertainty models with proven statistical guarantees that do not assume any shape for the error distribution of the surrogate model. However, since the classic statistical guarantee offered by CP is given in terms of bounds for the marginal coverage, for small calibration set sizes (which are frequent in realistic surrogate modelling that aims to quantify error at different regions), the potentially strong dispersion of the coverage distribution around its average negatively impacts the relevance of the uncertainty model's statistical guarantee, often obtaining coverages below the expected value, resulting in a less applicable framework. After providing a gentle presentation of uncertainty quantification for surrogate models for machine learning practitioners, in this paper we bridge the gap by proposing a new statistical guarantee that offers probabilistic information for the coverage of a single conformal predictor. We show that the proposed framework converges to the standard solution offered by CP for large calibration set sizes and, unlike the classic guarantee, still offers relevant information about the coverage of a conformal predictor for small data sizes. We validate the methodology in a suite of examples, and implement an open access software solution that can be used alongside common conformal prediction libraries to obtain uncertainty models that fulfil the new guarantee.

cs.LG↗

arXiv2025

A Certifiable Machine Learning-Based Pipeline to Predict Fatigue Life of Aircraft Structures

Fatigue life prediction is essential in both the design and operational phases of any aircraft, and in this sense safety in the aerospace industry requires early detection of fatigue cracks to prevent in-flight failures. Robust and precise fatigue life predictors are thus essential to ensure safety. Traditional engineering methods, while reliable, are time consuming and involve complex workflows, including steps such as conducting several Finite Element Method (FEM) simulations, deriving the expected loading spectrum, and applying cycle counting techniques like peak-valley or rainflow counting. These steps often require collaboration between multiple teams and tools, added to the computational time and effort required to achieve fatigue life predictions. Machine learning (ML) offers a promising complement to traditional fatigue life estimation methods, enabling faster iterations and generalization, providing quick estimates that guide decisions alongside conventional simulations. In this paper, we present a ML-based pipeline that aims to estimate the fatigue life of different aircraft wing locations given the flight parameters of the different missions that the aircraft will be operating throughout its operational life. We validate the pipeline in a realistic use case of fatigue life estimation, yielding accurate predictions alongside a thorough statistical validation and uncertainty quantification. Our pipeline constitutes a complement to traditional methodologies by reducing the amount of costly simulations and, thereby, lowering the required computational and human resources.

cs.LG↗

arXiv2025

Transfer learning-enhanced deep reinforcement learning for aerodynamic airfoil optimisation subject to structural constraints

The main objective of this paper is to introduce a transfer learning-enhanced deep reinforcement learning (DRL) methodology that is able to optimise the geometry of any airfoil based on concomitant aerodynamic and structural integrity criteria. To showcase the method, we aim to maximise the lift-to-drag ratio $C_L/C_D$ while preserving the structural integrity of the airfoil -- as modelled by its maximum thickness -- and train the DRL agent using a list of different transfer learning (TL) strategies. The performance of the DRL agent is compared with Particle Swarm Optimisation (PSO), a traditional gradient-free optimisation method. Results indicate that DRL agents are able to perform purely aerodynamic and hybrid aerodynamic/structural shape optimisation, that the DRL approach outperforms PSO in terms of computational efficiency and aerodynamic improvement, and that the TL-enhanced DRL agent achieves performance comparable to the DRL one, while further saving substantial computational resources.

cs.LG↗

arXiv2025

Scalar embedding of temporal network trajectories

A temporal network -- a collection of snapshots recording the evolution of a network whose links appear and disappear dynamically -- can be interpreted as a trajectory in graph space. In order to characterize the complex dynamics of such trajectory via the tools of time series analysis and signal processing, it is sensible to preprocess the trajectory by embedding it in a low-dimensional Euclidean space. Here we argue that, rather than the topological structure of each network snapshot, the main property of the trajectory that needs to be preserved in the embedding is the relative graph distance between snapshots. This idea naturally leads to dimensionality reduction approaches that explicitly consider relative distances, such as Multidimensional Scaling (MDS) or identifying the distance matrix as a feature matrix in which to perform Principal Component Analysis (PCA). This paper provides a comprehensible methodology that illustrates this approach. Its application to a suite of generative network trajectory models and empirical data certify that nontrivial dynamical properties of the network trajectories are preserved already in their scalar embeddings, what enables the possibility of performing time series analysis in temporal networks.

physics.soc-ph↗

arXiv2025

Predictability of temporal network dynamics in normal ageing and brain pathology

Spontaneous brain activity generically displays transient spatiotemporal coherent structures, which can selectively be affected in various neurological and psychiatric pathologies. Here we model the full brain's electroencephalographic activity as a high-dimensional functional network performing a trajectory in a latent graph phase space. This approach allows us to investigate the orbital stability of brain's activity and in particular its short-term predictability. We do this by constructing a non-parametric statistic quantifying the expansion of initially close functional network trajectories. We apply the method to cohorts of healthy ageing individuals, and patients previously diagnosed with Parkinson's or Alzheimer's disease. Results not only characterise brain dynamics from a new angle, but further show that functional network predictability varies in a marked scale-dependent way across healthy controls and patient groups. The path towards both pathologies is markedly different. Furthermore, healthy ageing's predictability appears to strongly differ from that of Parkinson's disease, but much less from that of patients with Alzheimer's disease.

q-bio.NC↗

arXiv2024

Characterising the dynamics of unlabelled temporal networks

Networks model the architecture backbone of complex systems. The backbone itself can change over time leading to what is called `temporal networks'. Interpreting temporal networks as trajectories in graph space of a latent graph dynamics has recently enabled the extension of concepts and tools from dynamical systems and time series to networks. Here we address temporal networks with unlabelled nodes, a case that has received relatively little attention so far. Situations in which node labelling cannot be tracked over time often emerge in practice due to technical challenges, or privacy constraints. In unlabelled temporal networks there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabelled network trajectories is nontrivial. We here exploit graph invariants to extend to the unlabelled setting network-dynamical quantifiers of linear correlations and dynamical instability. In particular, we focus on autocorrelation functions and the sensitive dependence on initial conditions. We show with synthetic graph dynamics that the measures are capable of recovering and estimating these dynamical fingerprints even when node labels are unavailable. We also validate the methods for some empirical temporal networks with removed node labels.

physics.soc-ph↗

arXiv2024

Towards certification: A complete statistical validation pipeline for supervised learning in industry

Methods of Machine and Deep Learning are gradually being integrated into industrial operations, albeit at different speeds for different types of industries. The aerospace and aeronautical industries have recently developed a roadmap for concepts of design assurance and integration of neural network-related technologies in the aeronautical sector. This paper aims to contribute to this paradigm of AI-based certification in the context of supervised learning, by outlining a complete validation pipeline that integrates deep learning, optimization and statistical methods. This pipeline is composed by a directed graphical model of ten steps. Each of these steps is addressed by a merging key concepts from different contributing disciplines (from machine learning or optimization to statistics) and adapting them to an industrial scenario, as well as by developing computationally efficient algorithmic solutions. We illustrate the application of this pipeline in a realistic supervised problem arising in aerostructural design: predicting the likelikood of different stress-related failure modes during different airflight maneuvers based on a (large) set of features characterising the aircraft internal loads and geometric parameters.

cs.LG↗

arXiv2024

Dynamical stability and chaos in artificial neural network trajectories along training

The process of training an artificial neural network involves iteratively adapting its parameters so as to minimize the error of the network's prediction, when confronted with a learning task. This iterative change can be naturally interpreted as a trajectory in network space -- a time series of networks -- and thus the training algorithm (e.g. gradient descent optimization of a suitable loss function) can be interpreted as a dynamical system in graph space. In order to illustrate this interpretation, here we study the dynamical properties of this process by analyzing through this lens the network trajectories of a shallow neural network, and its evolution through learning a simple classification task. We systematically consider different ranges of the learning rate and explore both the dynamical and orbital stability of the resulting network trajectories, finding hints of regular and chaotic behavior depending on the learning rate regime. Our findings are put in contrast to common wisdom on convergence properties of neural networks and dynamical systems theory. This work also contributes to the cross-fertilization of ideas between dynamical systems theory, network theory and machine learning

cs.LG↗

arXiv2023

An effective theory of collective deep learning

Unraveling the emergence of collective learning in systems of coupled artificial neural networks points to broader implications for machine learning, neuroscience, and society. Here we introduce a minimal model that condenses several recent decentralized algorithms by considering a competition between two terms: the local learning dynamics in the parameters of each neural network unit, and a diffusive coupling among units that tends to homogenize the parameters of the ensemble. We derive an effective theory for linear networks to show that the coarse-grained behavior of our system is equivalent to a deformed Ginzburg-Landau model with quenched disorder. This framework predicts depth-dependent disorder-order-disorder phase transitions in the parameters' solutions that reveal a depth-delayed onset of a collective learning phase and a low-rank microscopic learning path. We validate the theory in coupled ensembles of realistic neural networks trained on the MNIST dataset under privacy constraints. Interestingly, experiments confirm that individual networks -- trained on private data -- can fully generalize to unseen data classes when the collective learning phase emerges. Our work establishes the physics of collective learning and contributes to the mechanistic interpretability of deep learning in decentralized settings.

physics.soc-ph↗

arXiv2023

Network bypasses sustain complexity

Real-world networks are neither regular nor random, a fact elegantly explained by mechanisms such as the Watts-Strogatz or the Barabasi-Albert models, among others. Both mechanisms naturally create shortcuts and hubs, which while enhancing network's connectivity, also might yield several undesired navigational effects: they tend to be overused during geodesic navigational processes -- making the networks fragile -- and provide suboptimal routes for diffusive-like navigation. Why, then, networks with complex topologies are ubiquitous? Here we unveil that these models also entropically generate network bypasses: alternative routes to shortest paths which are topologically longer but easier to navigate. We develop a mathematical theory that elucidates the emergence and consolidation of network bypasses and measure their navigability gain. We apply our theory to a wide range of real-world networks and find that they sustain complexity by different amounts of network bypasses. At the top of this complexity ranking we found the human brain, which points out the importance of these results to understand the plasticity of complex systems.

physics.soc-ph↗