Peter van Heijster

Eva van Tegelen, Taniya Kapoor, George A. K. van Voorn et al.

Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications. In practical applications, variations in physical parameters induce distribution shifts between training and prediction regimes, making extrapolation a central challenge. As a result, the way parameters are incorporated into neural operator models plays a key role in their ability to generalize, particularly when state and parameter representations are entangled. In this work, we introduce the Late Fusion Neural Operator, an architecture that disentangles learning state dynamics from parameter effects, improving predictive performance both within and beyond the training distribution. Our approach combines neural operators for learning latent state representations with sparse regression to incorporate parameter information in a structured manner. Across four benchmark PDEs including advection, Burgers, and both 1D and 2D reaction-diffusion equations, the proposed method consistently outperforms Fourier Neural Operator and CAPE-FNO. Late Fusion Neural Operators achieve consistently the best performance in all experiments, with an average RMSE reduction of 72.9% in-domain and 71.8% out-domain compared to the second-best method. These results demonstrate strong generalization across both in-domain and out-domain parameter regimes.

Eva van Tegelen, George van Voorn, Ioannis Athanasiadis et al.

Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data, most studies remain limited as they exclusively focus on discrete-time methods and local bifurcations. To address these limitations, we use Neural Ordinary Differential Equations which provide a data-driven framework for learning system dynamics. Our results show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from time-series data by learning parameter-dependent vector fields. Notably, we demonstrate that Neural Ordinary Differential Equations can forecast bifurcations even beyond the parameter regions represented in the training data. We demonstrate our approach on three test cases: the Lorenz system transitioning from non-chaotic to chaotic behaviour, the Rössler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.