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Artificial intelligence, and in particular neural networks, offers a promising alternative. These models can learn patterns from data and make predictions much faster than traditional simulation techniques. This makes them attractive for applications where fast or repeated simulations are needed, for example in design optimization or uncertainty analysis.

However, as these methods are increasingly used in scientific settings, important questions arise about their reliability. Standard neural networks are not designed with physical laws in mind, and may therefore produce predictions that violate fundamental principles such as conservation of energy. While these errors may not always be immediately visible, they can accumulate over time and lead to unreliable long-term predictions.

PhD researcher set out to address these challenges by investigating how neural networks can be designed to respect the laws of physics. His research focuses on developing structure-preserving neural networks that incorporate physical laws directly into their design, with the goal of improving the reliability and long-term stability of AI-based simulations. He defended his PhD thesis at the Department of Mathematics and Computer Science on Wednesday, May 20, 2026.

Embedding physics into neural networks

Horn’s research focused on a class of physical systems known as . These systems describe many important phenomena in nature, including planetary motion and molecular dynamics, and are characterized by the conservation of energy.

Instead of using generic neural networks, Horn developed models that are explicitly designed to respect these physical laws. These structure-preserving neural networks build on ideas from classical numerical methods used to solve physical systems in a stable way, in particular symplectic integrators, which preserve important physical properties over long simulations.

By incorporating similar structure into neural networks, the goal is to ensure that learned models remain physically consistent, even when used for long-term predictions.

New neural network frameworks for physical systems

A central contribution of this research is the development of Generalized Hamiltonian Neural Networks (GHNNs). This framework unifies several earlier approaches into a single, more flexible architecture for learning Hamiltonian systems.

Experiments showed that this architecture achieved strong predictive performance compared to both standard neural networks and earlier structure-preserving methods. The results also highlighted an important trade-off: when sufficient training data is available and predictions remain close to that data, simpler neural networks can perform similarly well.

However, differences become clear when models are used beyond the range of the training data. In these cases, structure-preserving neural networks were significantly more reliable.

Extension to parametric systems

The research further extended this framework to parametric Hamiltonian systems, where physical systems can vary continuously, for example a pendulum with different lengths. For this broader class of problems, Horn developed Parametric Generalized Hamiltonian Neural Networks (PGHNNs).

For this model class, the thesis established a universal approximation theorem, showing that the networks can represent a wide range of physical systems.