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Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise representation of processes characterised by non-local and memory-dependent behaviours. This property is useful in systems where variables do not respond to changes instantaneously, but instead exhibit a strong memory of past interactions. Having this in mind, and drawing inspiration from Neural Ordinary Differential Equations (Neural ODEs), we propose the Neural FDE, a novel deep neural network architecture that adjusts a FDE to the dynamics of data. This work provides a comprehensive overview of the numerical method employed in Neural FDEs and the Neural FDE architecture. The numerical outcomes suggest that, despite being more computationally demanding, the Neural FDE may outperform the Neural ODE in modelling systems with memory or dependencies on past states, and it can effectively be applied to learn more intricate dynamical systems.

Real-world systems are often formulated as constrained optimization problems. Techniques to incorporate constraints into Neural Networks (NN), such as Neural Ordinary Differential Equations (Neural ODEs), have been used. However, these introduce hyperparameters that require manual tuning through trial and error, raising doubts about the successful incorporation of constraints into the generated model. This paper describes in detail the two-stage training method for Neural ODEs, a simple, effective, and penalty parameter-free approach to model constrained systems. In this approach the constrained optimization problem is rewritten as two unconstrained sub-problems that are solved in two stages. The first stage aims at finding feasible NN parameters by minimizing a measure of constraints violation. The second stage aims to find the optimal NN parameters by minimizing the loss function while keeping inside the feasible region. We experimentally demonstrate that our method produces models that satisfy the constraints and also improves their predictive performance. Thus, ensuring compliance with critical system properties and also contributing to reducing data quantity requirements. Furthermore, we show that the proposed method improves the convergence to an optimal solution and improves the explainability of Neural ODE models. Our proposed two-stage training method can be used with any NN architectures.

Hydropower plants play a pivotal role in advancing clean and sustainable energy production, contributing significantly to the global transition towards renewable energy sources. However, hydropower plants are currently perceived both positively as sources of renewable energy and negatively as disruptors of ecosystems. In this work, we highlight the overlooked potential of using hydropower plant as protectors of ecosystems by using adaptive ecological discharges. To advocate for this perspective, we propose using a neural network to predict the minimum ecological discharge value at each desired time. Additionally, we present a novel framework that seamlessly integrates it into hydropower management software, taking advantage of the well-established approach of using traditional constrained optimisation algorithms. This novel approach not only protects the ecosystems from climate change but also contributes to potentially increase the electricity production.

Hydropower plants are crucial for stable renewable energy and serve as vital water sources for sustainable agriculture. However, it is essential to assess the current water management practices associated with hydropower plant management software. A key concern is the potential conflict between electricity generation and agricultural water needs. Prioritising water for electricity generation can reduce irrigation availability in agriculture during crucial periods like droughts, impacting crop yields and regional food security. Coordination between electricity and agricultural water allocation is necessary to ensure optimal and environmentally sound practices. Neural networks have become valuable tools for hydropower plant management, but their black-box nature raises concerns about transparency in decision making. Additionally, current approaches often do not take advantage of their potential to create a system that effectively balances water allocation. This work is a call for attention and highlights the potential risks of deploying neural network-based hydropower plant management software without proper scrutiny and control. To address these concerns, we propose the adoption of the Agriculture Conscious Hydropower Plant Management framework, aiming to maximise electricity production while prioritising stable irrigation for agriculture. We also advocate reevaluating government-imposed minimum water guidelines for irrigation to ensure flexibility and effective water allocation. Additionally, we suggest a set of regulatory measures to promote model transparency and robustness, certifying software that makes conscious and intelligent water allocation decisions, ultimately safeguarding agriculture from undue strain during droughts.

The continuous dynamics of natural systems has been effectively modelled using Neural Ordinary Differential Equations (Neural ODEs). However, for accurate and meaningful predictions, it is crucial that the models follow the underlying rules or laws that govern these systems. In this work, we propose a self-adaptive penalty algorithm for Neural ODEs to enable modelling of constrained natural systems. The proposed self-adaptive penalty function can dynamically adjust the penalty parameters. The explicit introduction of prior knowledge helps to increase the interpretability of Neural ODE -based models. We validate the proposed approach by modelling three natural systems with prior knowledge constraints: population growth, chemical reaction evolution, and damped harmonic oscillator motion. The numerical experiments and a comparison with other penalty Neural ODE approaches and \emph{vanilla} Neural ODE, demonstrate the effectiveness of the proposed self-adaptive penalty algorithm for Neural ODEs in modelling constrained natural systems. Moreover, the self-adaptive penalty approach provides more accurate and robust models with reliable and meaningful predictions.

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent Neural Networks (RNNs), RNNs have been generalised to have continuous-time hidden dynamics defined by a Neural Ordinary Differential Equation (Neural ODE), leading to the ODE-RNN model. Another approach that provides a better modelling is that of the Latent ODE model, which constructs a continuous-time model where a latent state is defined at all times. The Latent ODE model uses a standard RNN as the encoder and a Neural ODE as the decoder. However, since the RNN encoder leads to difficulties with missing data and ill-defined latent variables, a Latent ODE-RNN model has recently been proposed that uses a ODE-RNN model as the encoder instead. Both the Latent ODE and Latent ODE-RNN models are difficult to train due to the vanishing and exploding gradients problem. To overcome this problem, the main contribution of this paper is to propose and illustrate a new model based on a new Latent ODE using an ODE-LSTM (Long Short-Term Memory) network as an encoder -- the Latent ODE-LSTM model. To limit the growth of the gradients the Norm Gradient Clipping strategy was embedded on the Latent ODE-LSTM model. The performance evaluation of the new Latent ODE-LSTM (with and without Norm Gradient Clipping) for modelling CTS with regular and irregular sampling rates is then demonstrated. Numerical experiments show that the new Latent ODE-LSTM performs better than Latent ODE-RNNs and can avoid the vanishing and exploding gradients during training.

This work introduces Neural Chronos Ordinary Differential Equations (Neural CODE), a deep neural network architecture that fits a continuous-time ODE dynamics for predicting the chronology of a system both forward and backward in time. To train the model, we solve the ODE as an initial value problem and a final value problem, similar to Neural ODEs. We also explore two approaches to combining Neural CODE with Recurrent Neural Networks by replacing Neural ODE with Neural CODE (CODE-RNN), and incorporating a bidirectional RNN for full information flow in both time directions (CODE-BiRNN), and variants with other update cells namely GRU and LSTM: CODE-GRU, CODE-BiGRU, CODE-LSTM, CODE-BiLSTM. Experimental results demonstrate that Neural CODE outperforms Neural ODE in learning the dynamics of a spiral forward and backward in time, even with sparser data. We also compare the performance of CODE-RNN/-GRU/-LSTM and CODE-BiRNN/-BiGRU/-BiLSTM against ODE-RNN/-GRU/-LSTM on three real-life time series data tasks: imputation of missing data for lower and higher dimensional data, and forward and backward extrapolation with shorter and longer time horizons. Our findings show that the proposed architectures converge faster, with CODE-BiRNN/-BiGRU/-BiLSTM consistently outperforming the other architectures on all tasks., Comment: Under review at journal

An analysis is performed of the liquid-gas phase transition of nuclear matter obtained from different versions of scalar derivate coupling suggested by Zimanyi and Moszkowski (ZM) and the results are compared with those obtained from the Walecka model. We present the phase diagram for the models and one of them, the ZM3 model, has the lowest critical temperature $T_c=13.6$ MeV with the lowest critical density $\rho_c=0.037$ f$m^{-3}$ and pressure $p_c=0.157$ MeV f$m^{-3}$. These results are in accord with recent observations from energetic heavy-ion collisions, which suggest a small liquid-gas phase region., Comment: 10 pages latex file, 2 tables and 7 Postscript figures