Mapping lakes and learning Hamiltonians

By Håkon Noren

While the tiny boat was immersed in darkness on the largest freshwater lake in Norway, Mjøsa, an underwater remotely operated vehicle (ROV) made its way down the water column. It was heading towards a particular coordinate, upon where HUGIN, an autonomous underwater vehicle (AUV) had spotted the tail fins of bombs dumped up to the 1970s with its sonar imaging technology. If the research team could only avoid tangling the ROV tether cable to the tail fin of the bombs, the team could return with a set of spectacular images.

Left: ROV diving in the night. Right: bomb tail fin 417 meters below the surface.

This is not an excerpt from a science fiction novel, but a report from the research project “Mission Mjøsa”, aiming, among other objectives, to map the bed of the lake where massive volumes of ammunition and explosives were dumped during the last century. In addition, there is a significant archeological interest in the sonar images from the HUGIN AUV [1], which has already discovered a shipwreck that may originate from the year 1300 – 1850 AD [2].

Left: The AUV HUGIN. Right: Sonar image of a cluster of crates with ammunition.

Ensuring the safe operation of autonomous vehicles, represents an interesting challenge in fields as cybernetics as well as mathematics and the study of dynamical systems. The project “DynNoise: Learning dynamical systems from noisy data” is a collaboration between NTNU, SINTEF, FFI and the research project AMOS. One of the crew members on the tiny boat on Mjøsa, Håkon Noren, writes his Ph.D. thesis on DynNoise at the department of Mathematical sciences at NTNU under the supervision of Professor Elena Celledoni (NTNU) and Professor Asgeir Sørensen (AMOS, NTNU). 

By combining methods from deep learning and theory from classical mechanics and geometric numerical integration, Håkon develops algorithms to learn a function, (the Hamiltonian) that describes the energy of a dynamical system, from observations. [3] Then, the learned Hamiltonian could be used to simulate the dynamics of the system. Such systems assume that the energy is preserved, but generalizations called Port-Hamiltonian systems could be used to include dissipation of energy and so-called forcing terms (describing external forces on the system). If you can collect accurate observations of a system to learn an accurate representation of the (Port-) Hamiltonian function, this could be used to control the system. 

Learning the double pendulum Hamiltonian. Left: Sample of data points. Right: simulated trajectories.

In many cases it is possible to derive analytical expressions for the dynamical system one aims to control, such as a drone or an AUV. However, when the interaction of the vehicle with the environment is more complicated, consider for instance an autonomous surface vehicle on a lake with waves, this is much harder. In this case, a representation of the system learned from data might be better and yield more accurate control.



Mathematics in Technology – Strategic Research Area
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