In recent years, nonlinear dynamic system identification using artificial neural networks has garnered attention due to its manifold potential applications in virtually all branches of science and engineering. However, purely data-driven approaches often struggle with extrapolation and may yield physically implausible forecasts. Furthermore, the learned dynamics can exhibit instabilities, making it difficult to apply such models safely and robustly. This article proposes stable port-Hamiltonian neural networks, a machine learning architecture that incorporates the physical biases of energy conservation or dissipation while guaranteeing global Lyapunov stability of the learned dynamics. Evaluations with illustrative examples and real-world measurement data demonstrate the model's ability to generalize from sparse data, outperforming purely data-driven approaches and avoiding instability issues. In addition, the model's potential for data-driven surrogate modeling is highlighted in application to multi-physics simulation data.

The success of Large Language Models (LLMs), e.g., ChatGPT, is witnessed by their planetary popularity, their capability of human-like communication, and also by their steadily improved reasoning performance. However, it remains unclear whether LLMs reason. It is an open problem how traditional neural networks can be qualitatively extended to go beyond the statistic paradigm and achieve high-level cognition. Here, we present a novel qualitative extension by generalising computational building blocks from vectors to spheres. We propose Sphere Neural Networks (SphNNs) for human-like reasoning through model construction and inspection, and develop SphNN for syllogistic reasoning, a microcosm of human rationality. SphNN is a hierarchical neuro-symbolic Kolmogorov-Arnold geometric GNN, and uses a neuro-symbolic transition map of neighbourhood spatial relations to transform the current sphere configuration towards the target. SphNN is the first neural model that can determine the validity of long-chained syllogistic reasoning in one epoch without training data, with the worst computational complexity of O(N). SphNN can evolve into various types of reasoning, such as spatio-temporal reasoning, logical reasoning with negation and disjunction, event reasoning, neuro-symbolic unification, and humour understanding (the highest level of cognition). All these suggest a new kind of Herbert A. Simon's scissors with two neural blades. SphNNs will tremendously enhance interdisciplinary collaborations to develop the two neural blades and realise deterministic neural reasoning and human-bounded rationality and elevate LLMs to reliable psychological AI. This work suggests that the non-zero radii of spheres are the missing components that prevent traditional deep-learning systems from reaching the realm of rational reasoning and cause LLMs to be trapped in the swamp of hallucination.

Dataset augmentation is a common way to deal with small datasets; Chemistry datasets are often small. Spherical convolutional neural networks (SphNNs) and Icosahedral neural networks (IcoNNs) are a type of geometric machine learning algorithm that maintains rotational symmetry. Molecular structure has rotational invariance and is inherently 3-D, and thus we need 3-D encoding methods to input molecular structure into machine learning. In this paper I present Icospherical Chemical Objects (ICOs) that enable the encoding of 3-D data in a rotationally invariant way which works with spherical or icosahedral neural networks and allows for dataset augmentation. I demonstrate the ICO featurisation method on the following tasks: predicting general molecular properties, predicting solubility of drug like molecules and the protein binding problem and find that ICO and SphNNs perform well on all problems.