Estimation of parameters in differential equation models can be achieved by applying learning algorithms to quantitative time-series data. However, sometimes it is only possible to measure qualitative changes of a system in response to a controlled condition. In dynamical systems theory, such change points are known as bifurcations and lie on a function of the controlled condition called the bifurcation diagram. In this work, we propose a gradient-based approach for inferring the parameters of differential equations that produce a user-specified bifurcation diagram. The cost function contains an error term that is minimal when the model bifurcations match the specified targets and a bifurcation measure which has gradients that push optimisers towards bifurcating parameter regimes. The gradients can be computed without the need to differentiate through the operations of the solver that was used to compute the diagram. We demonstrate parameter inference with minimal models which explore the space of saddle-node and pitchfork diagrams and the genetic toggle switch from synthetic biology. Furthermore, the cost landscape allows us to organise models in terms of topological and geometric equivalence.

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.

Artificial Intelligence has advanced significantly in recent years thanks to innovations in the design and training of artificial neural networks (ANNs). Despite these advancements, we still understand relatively little about how elementary forms of ANNs learn, fail to learn, and generate false information without the intent to deceive, a phenomenon known as `confabulation'. To provide some foundational insight, in this paper we analyse how confabulation occurs in reservoir computers (RCs): a dynamical system in the form of an ANN. RCs are particularly useful to study as they are known to confabulate in a well-defined way: when RCs are trained to reconstruct the dynamics of a given attractor, they sometimes construct an attractor that they were not trained to construct, a so-called `untrained attractor' (UA). This paper sheds light on the role played by UAs when reconstruction fails and their influence when modelling transitions between reconstructed attractors. Based on our results, we conclude that UAs are an intrinsic feature of learning systems whose state spaces are bounded, and that this means of confabulation may be present in systems beyond RCs.

This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high-dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory. Numerical experiments are conducted on the discrete Allen-Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high-dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.

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.

Machine learning (ML) models can be effective for forecasting the dynamics of unknown systems from time-series data, but they often require large amounts of data and struggle to generalize across systems with varying dynamics. Combined, these issues make forecasting from short time series particularly challenging. To address this problem, we introduce Meta-learning for Tailored Forecasting from Related Time Series (METAFORS), which uses related systems with longer time-series data to supplement limited data from the system of interest. By leveraging a library of models trained on related systems, METAFORS builds tailored models to forecast system evolution with limited data. Using a reservoir computing implementation and testing on simulated chaotic systems, we demonstrate METAFORS' ability to predict both short-term dynamics and long-term statistics, even when test and related systems exhibit significantly different behaviors and the available data are scarce, highlighting its robustness and versatility in data-limited scenarios.