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.

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.

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.

Leveraging chaos 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 and other hyperparameters, and showcase the emergent, constructive role of transient chaotic dynamics in the training of artificial neural networks.

Aerodynamic and structural airfoil shape optimisation via Transfer Learning-enhanced Deep Reinforcement Learning

The main objective of this paper is to introduce a transfer learning-enhanced, multi-objective, deep reinforcement learning (DRL) methodology that is able to optimise the geometry of any airfoil based on concomitant aerodynamic and structural 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 multi-objective shape optimisation, that the DRL approach outperforms PSO in terms of computational efficiency and shape optimisation performance, and that the TL-enhanced DRL agent achieves performance comparable to the DRL one, while further saving substantial computational resources.

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.

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

An effective theory of collective deep learning

Unraveling the emergence of collective learning in systems of coupled artificial neural networks is an endeavor with broader implications for physics, 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 the coarse-grained behavior of our model via an effective theory for linear networks that we show is analogous 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 the onset of a collective learning phase, along with a depth-induced delay of the critical point and a robust shape of the microscopic learning path. We validate our theory in realistic ensembles of coupled nonlinear networks trained in the MNIST dataset under privacy constraints. Interestingly, experiments confirm that individual networks -- trained only with private data -- can fully generalize to unseen data classes when the collective learning phase emerges. Our work elucidates the physics of collective learning and contributes to the mechanistic interpretability of deep learning in decentralized settings.

Predicting e-commerce customer conversion from minimal temporal patterns on symbolized clickstream trajectories

Knowing if a user is a buyer or window shopper solely based on clickstream data is of crucial importance for e-commerce platforms seeking to implement real-time accurate NBA (next best action) policies. However, due to the low frequency of conversion events and the noisiness of browsing data, classifying user sessions is very challenging. In this paper, we address the clickstream classification problem in the eCommerce industry and present three major contributions to the burgeoning field of AI-for-retail: first, we collected, normalized and prepared a novel dataset of live shopping sessions from a major European e-commerce website; second, we use the dataset to test in a controlled environment strong baselines and SOTA models from the literature; finally, we propose a new discriminative neural model that outperforms neural architectures recently proposed at Rakuten labs.

Prediction is very hard, especially about conversion. Predicting user purchases from clickstream data in fashion e-commerce

Knowing if a user is a buyer vs window shopper solely based on clickstream data is of crucial importance for ecommerce platforms seeking to implement real-time accurate NBA (next best action) policies. However, due to the low frequency of conversion events and the noisiness of browsing data, classifying user sessions is very challenging. In this paper, we address the clickstream classification problem in the fashion industry and present three major contributions to the burgeoning field of AI in fashion: first, we collected, normalized and prepared a novel dataset of live shopping sessions from a major European e-commerce fashion website; second, we use the dataset to test in a controlled environment strong baselines and SOTA models from the literature; finally, we propose a new discriminative neural model that outperforms neural architectures recently proposed at Rakuten labs.