Proceedings of the 2nd ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications" Held in University of Bologna - The Engineering School, Bologna, Italy on 26 October 2025 Published as Volume 277 by the Proceedings of Machine Learning Research on 07 October 2025. Volume Edited by: Cecı́lia Coelho Bernd Zimmering M. Fernanda P. Costa Luı́s L. Ferrás Oliver Niggemann Series Editors: Neil D. Lawrence https://proceedings.mlr.press/v277/ Sun, 05 Jul 2026 15:02:22 +0000 Sun, 05 Jul 2026 15:02:22 +0000 Jekyll v3.10.0 Many conservative physical systems can be described using the Hamiltonian formalism. A notable example are the Vlasov-Poisson equations, a set of partial differential equations that govern the time evolution of a phase-space density function representing collisionless particles under a self-consistent potential. These equations play a central role in both plasma physics and cosmology. Due to the complexity of the potential involved, analytical solutions are rarely available, necessitating the use of numerical methods such as Particle-In-Cell. In this work, we introduce a novel approach based on Hamiltonian-informed Normalizing Flows, specifically a variant of Fixed-Kinetic Neural Hamiltonian Flows. Our method transforms an initial Gaussian distribution in phase space into the final distribution using a sequence of invertible, volume-preserving transformations derived from Hamiltonian dynamics. The model is trained on a dataset comprising initial and final states at a fixed time T, generated via numerical simulations. After training, the model enables fast sampling of the final distribution from any given initial state. Moreover, by automatically learning an interpretable physical potential, it can generalize to intermediate states not seen during training, offering insights into the system’s evolution across time. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/souveton25a.html https://proceedings.mlr.press/v277/souveton25a.html Neural ordinary differential equations offer an effective framework for modeling dynamical systems by learning a continuous-time vector field. However, they rely on the Markovian assumption-that future states depend only on the current state-which is often untrue in real-world scenarios where the dynamics may depend on the history of past states. This limitation becomes especially evident in settings involving the continuous control of complex systems with delays and memory effects. To capture historical dependencies, existing approaches often rely on recurrent neural network (RNN)-based encoders, which are inherently discrete and struggle with continuous modeling. In addition, they may exhibit poor training behavior. In this work, we investigate the use of the signature transform as an encoder for learning non-Markovian dynamics in a continuous-time setting. The signature transform offers a continuous-time alternative with strong theoretical foundations and proven efficiency in summarizing multidimensional information in time. We integrate a signature-based encoding scheme into encoder-decoder dynamics models and demonstrate that it outperforms RNN-based alternatives in test performance on synthetic benchmarks. The code is available at : https://github.com/eliottprdlx/Signature-Encoders-For-Dynamics-Learning.git. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/pradeleix25a.html https://proceedings.mlr.press/v277/pradeleix25a.html This paper proposes Hamiltonian Learning, a novel unified framework for learning with neural networks "over time", i.e., from a possibly infinite stream of data, in an online manner, without having access to future information. In this paper, the problem of learning over time is rethought from scratch, leveraging tools from optimal control theory, which yield a unifying view of the temporal dynamics of neural computations and learning. Hamiltonian Learning is based on differential equations that: (i) can be integrated without the need of external software solvers; (ii) generalize the well-established notion of gradient-based learning in feed-forward and recurrent networks; (iii) open to novel perspectives. The proposed framework is showcased by experimentally proving how it can recover gradient-based learning, comparing it to out-of-the box optimizers, and describing how it is flexible enough to switch from fully-local to partially/non-local computational schemes, possibly distributed over multiple devices, and BackPropagation without storing activations. Hamiltonian Learning is easy to implement and can help researches approach in a principled and innovative manner the problem of learning over time. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/melacci25a.html https://proceedings.mlr.press/v277/melacci25a.html Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorNet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorNet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/dulny25a.html https://proceedings.mlr.press/v277/dulny25a.html Current flow based generative speech enhancement methods learn curved probability paths which model a mapping between clean and noisy speech. Despite impressive performance, the implications of curved probability paths are unknown. Methods such as Schr\"{}odinger bridges focus on curved paths, where time dependent gradients and variance do not promote straight paths. Findings in machine learning research suggest that straight paths, such as conditional flow matching, are easier to train and offer better generalisation. In this paper we quantify the effect of path straightness on speech enhancement quality. We report experiments with the Schrödinger bridge, where we show that certain configurations lead to straighter paths. Conversely, we propose independent conditional flow matching for speech enhancement, which models straight paths between noisy and clean speech. We demonstrate empirically that a time independent variance has a greater effect on sample quality than the gradient. Although conditional flow matching improves several speech quality metrics, it requires multiple inference steps. We rectify this with a one step solution by inferring the trained flow based model as if it was directly predictive. Our work suggests that straighter time independent probability paths improve generative speech enhancement over curved time dependent paths. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/cross25a.html https://proceedings.mlr.press/v277/cross25a.html The DeepONet, based on the Universal Approximation Theorem for Operators (UATO), made a significant impact on Deep Learning research due to its ability to learn maps between function spaces instead of between vector spaces like traditional Neural Networks. However, DeepONets are computationally expensive to train. To address this we introduce the ExtremONet: an Extreme-Learning-Machine-based variation of the DeepONet, capable of one-step learning maps between function spaces. We show that ExtremONets approach DeepONets in training error whilst displaying lower generalization error, and training between two to four orders of magnitude faster on small datasets. Our work represents an important step towards efficient dynamical modeling using Machine Learning. We conclude our analysis with an exploration of the ExtremONet’s Out-of-Distribution Generalization capabilities. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/beysen25a.html https://proceedings.mlr.press/v277/beysen25a.html Accurate air pollution forecasting is critical for public health and environmental policy, particularly in densely populated regions like the Netherlands. This work introduces a physics-informed graph neural network (PI-GNN) framework for urban nitrogen dioxide (NO2) forecasting, which integrates domain-specific physical constraints into graph-based deep learning models. By combining spatial and temporal learning with physical knowledge, the proposed physics-informed graph convolutional network with gated recurrent units significantly outperforms purely data-driven recurrent and graph neural networks in terms of accuracy, generalizability, and environmental efficiency. Moreover, physics-informed models demonstrated progressively better relative performance over purely data-driven models in conditions with scarce data. Tue, 07 Oct 2025 00:00:00 +0000 https://proceedings.mlr.press/v277/assiotis25a.html https://proceedings.mlr.press/v277/assiotis25a.html