The idea of physics informed learning

Economical or biological models, and models of physical systems are described by differential equations. In practice, these governing equations are often either unknown or only partial information is available. Using machine learning, missing knowledge about models can be learned from observational data. However, it appears natural to incorporate the prior knowledge that we have as this will make a learned model more reliable and reduce data requirements.

In our current research, we develop new methods to incorporate prior knowledge about the validity of fundamental physical principles into machine learned models of dynamical systems.

Hamilton’s Principle and Machine Learning

Hamilton’s principle is one of the most fundamental principle in physics. Incorporating the principle into data-driven models of dynamical systems guarantees that its motions share important qualitative properties with the real system, such as energy or momentum conservation. The principle states that motions of a dynamical systems extremise an action functional. Learning the action functional instead of the governing equations or its solutions makes sure that Hamilton’s principle applies to the motions of the learned model as well. The learned model is, therefore, guaranteed to behave more physically and exhibit structural properties such as energy conservation in case of autonomous systems. More generally, the powerful Noether theorem relating symmetries and conserved quantities applies to the learned model as well. Once the action functional has been identified, highly developed numerical algorithms are available to discretise and compute numerical solutions to the system, preserving Hamilton’s principle under discretisation.

Research Questions

In our current research we tackle questions such as

• an end-to-end optimisation of discovery and simulation of models of dynamical systems. Using backward error analysis of numerical methods, for instance, we can compensate for discretisation errors in the training data as well as in the numerical discretisation. This allow for efficient integrations with large time-steps, extremely high energy accuracy, and excellent long-term behaviour.

• We explore clever ways to incorporate symmetries into learning processes using neural networks as well as Gaussian Processes. Thanks to Noether’s theorem, this guarantees the presence of conservation laws which improve the model’s long-term behaviour.

• We develop frameworks to discover symmetries and conserved quantities in dynamical systems employing tools from structure preserving numerical integration and Differential Geometry.