Coelho, Cecília

Forecasting the behaviour of industrial robots, power grids or pandemics under changing external inputs requires accurate dynamical models that can adapt to varying signals and capture long-term effects such as delays or memory. While recent neural approaches address some of these challenges individually, their reliance on computationally intensive solvers and their black-box nature limit their practical utility. In this work, we propose Laplace-Net, a decoupled, solver-free neural framework for learning forced and delay-aware dynamical systems. It uses the Laplace transform to (i) bypass computationally intensive solvers, (ii) enable the learning of delays and memory effects and (iii) decompose each system into interpretable control-theoretic components. Laplace-Net also enhances transferability, as its modular structure allows for targeted re-training of individual components to new system setups or environments. Experimental results on eight benchmark datasets–including linear, nonlinear and delayed systems–demonstrate the method’s improved accuracy and robustness compared to state-of-the-art approaches, particularly in handling complex and previously unseen inputs.

Optimal control problems (OCPs) are essentials in various domains such as science, engineering, and industry, requiring the optimisation of control variables for dynamic systems, along with the corresponding state variables, that minimise a given performance index. Traditional methods for solving OCPs often rely on numerical techniques and can be computationally expensive when the discretisation grid or time horizon changes. In this work, we introduce a novel approach that leverages Neural Ordinary Differential Equations (Neural ODEs) to model the dynamics of control variables in OCPs. By embedding Neural ODEs within the optimisation problem, we effectively address the limitations of traditional methods, eliminating the need to re-solve the OCP under different discretisation schemes. We apply this method to a coastal ecosystem OCP, demonstrating its efficacy in solving the problem over a 50-year horizon and extending predictions up to 70 years without resolve the optimisation problem.

Neural Ordinary Differential Equations (NODEs) are well-established architectures that fit an ODE, modelled by a neural network (NN), to data, effectively modelling complex dynamical systems. Recently, Neural Fractional Differential Equations (NFDEs) were proposed, inspired by NODEs, to incorporate non-integer order differential equations, capturing memory effects and long-range dependencies. In this work, we present an optimised implementation of the NFDE solver, achieving up to 570 times faster computations and up to 79 times higher accuracy. Additionally, the solver supports efficient multidimensional computations and batch processing. Furthermore, we enhance the experimental design to ensure a fair comparison of NODEs and NFDEs by implementing rigorous hyperparameter tuning and using consistent numerical methods. Our results demonstrate that for systems exhibiting fractional dynamics, NFDEs significantly outperform NODEs, particularly in extrapolation tasks on unseen time horizons. Although NODEs can learn fractional dynamics when time is included as a feature to the NN, they encounter difficulties in extrapolation due to reliance on explicit time dependence. The code is available at https://github.com/zimmer-ing/Neural-FDE