DynNoise: Learning dynamical systems from noisy data

The computational paradigm underlies many of the advances that have helped define the modern world as we know it. In a text 1 that deals with how mathematics has contributed to development, it is described as a closed circuit between a physical problem and a mathematical representation of the problem. Measurements of physical quantities are used in a mathematical representation, and manipulation of the abstract problem provides a solution for how to drive the physical system to a desired position. Simulation and control of aircraft and satellites are good examples of how the computational paradigm is useful today. This paradigm is beneficial as it allows us to do things that were previously impossible, such as landing space probes on other planets. It provides increased accuracy, of which precision weapons are an uncomfortable example. Cost reduction is another positive outcome, as one can design vehicles with less air resistance or carry out optimal planning of timetables. It is therefore natural to think that if we could create mathematical representations of as much as possible, we as a society could perform new (useful) things, cheaper and more precisely. The challenge is that it is demanding to create a good mathematical representation of a physical problem. It requires people with specialized expertise who are both too few in number and expensive to maintain. However, one can come a step closer if one can take advantage of measurement data, which is becoming increasingly abundant. The next step is, of course, to use machine learning and AI to give us something that resembles as much as possible an abstract mathematical representation that can be used for simulation and control. We want both flexible algorithms that can find patterns in large datasets, but at the same time, we want to identify machine learning models that align with what we know to be true. Something we know to be true can be energy conservation in mechanical systems or that mass must be conserved in a fluid flow. Geometric numerical integration is precisely about developing algorithms for simulation that follow physical principles such as energy conservation, and combined with machine learning, we can solve the problem described above. Another challenge is that measurement data often contains noise. Thus, a good algorithm for identifying mathematical models must be sufficiently flexible to adapt to different measurement data, it must respect conservation laws, and be able to handle noise. A concrete example of this is given in this article 2, where we try to learn dynamical systems with energy conservation from noisy measurement data. Although the computational paradigm is starting to be well-tested, there is good reason to believe that there is much to gain from developing better computations applied to more areas in society. AI will accelerate this development, but first, we should teach this new type of intelligence what we can about physics. [1]: https://worksinprogress.co/issue/how-mathematics-built-the-modern-world/ [2]: https://arxiv.org/pdf/2306.03548

SINTEF Digital and the Norwegian Defence Research Establishment (FFI) are both involved as partners in the Centre for Autonomous Marine Operations and Systems (NTNU AMOS). We are seeking funding for two years of a PhD project within physics-informed machine learning, with applications to underwater robotics. The candidate will be employed at the Department of Mathematics at NTNU. In addition to AMOS, the PhD student will be connected to several other ongoing projects linked to this research area. Hybrid modeling combines data-driven and analytical modeling. This approach enables the simulation of dynamical systems from data, allowing physical principles and numerical analysis to inform and constrain deep learning models. In particular, Hamiltonian neural networks leverages the energy-preserving Hamiltonian formulation from classical mechanics to obtain a deep learning model that describes the dynamics of a physical system. Geometric numerical integration is a well-established field concerning numerical solutions that preserve geometric structure or follow first principles. Geometric integration and classical mechanics combine centuries of theoretical discoveries. Deep learning serves as an enabling technology allowing a range of complex systems to be modeled from data. A combination of these fields could trigger a revolution in mathematical modeling of dynamical systems. This PhD project aims at combining the mathematical rigor of geometric integration with the approximation capacity of neural networks, allowing neural networks to learn dynamical systems from data while preserving geometric structure. Furthermore, this project will explore how a learned system could support scientific discovery in underwater robotics and be applied in control of autonomous navigation vehicles.