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# IMod. Partial differential equations, statistics and data: An interdisciplinary approach to data-based modelling

**Alternative title:** IMod. Partielle differensiallikninger, statistikk og data: en tverrfaglig tilnærming til data-basert modellering

#### Awarded: NOK 6.5 mill.

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Project Manager:

Project Number:

325114

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Project Period:

2022 - 2028

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In this interdisciplinary project we combine knowledge from mathematics and statistics, with data from fields such as fluid mechanics and neuroscience, to build, analyse and apply models that can be used both to forecast and understand new situations.
Classical models for understanding the world – such as Newton’s law of motion in physics, or Einstein’s equations – typically reflect and are evaluated by few basic principles. They can be ’understood’, but fail to handle the more complex interactions in real-life settings (Newton’s law of motion does not aid you much in driving a car).
Modern models on the other side can take in vast amounts of data, process them and come up with quite good predictions for specific situations (such as driving). But the models are not ’understandable’, we cannot see their inner workings, and it is therefore hard to adjust them to similar but not identical situations, where they could have also been of great value.
The IMod project combines three types of specialists to try to bridge the gap between the two above types of models. Experimentalists and theorists in neuroscience and fluid mechanics can gather data in situations which they ’control’ and have concepts for. In some situations, there are models for these, whereas in some others the behaviour is yet to be understood. Statisticians and modellers then build models from this data using the knowledge of the situation, by combining both mathematical (exact) and statistical (noisy) techniques. These models, further, can be analysed in themselves, and compared with previous models and real-life situations to be evaluated and further improved.
The main goal of the project is to bring the fields of partial differential equations and statistical modelling closer to generate models that are data-based, but at the same time possible to analyse with mathematical tools. In neuroscience and fluid mechanics we aim to bring a new type of analysis into situations previously not understood.

This is an interdisciplinary proposal for building, analysing and testing a framework for data-based modelling built on the combination of partial differential equations and statistical modelling, and applied in particular to surface fluid mechanics and neuroscience. It is a 6-year project based on full funding with 7 PIs including the project manager, working closely together in 3 work packages to fulfil the proposal’s inter-disciplinary potential.
Today we can acquire data which only recently was far out of reach. The potential this creates for new scientific insight can only be fully realised once we have methods to make the data reveal its underlying mechanisms. We need simple and understandable mathematical models that can be analysed for both their qualitative and quantitative properties.
In fluid mechanics, neuroscience and other data-rich sciences there are well-structured systems, but also rich and complex behaviours that are not yet understood. In dealing with them, science is now standing in the crosswinds of top-down and bottom-up approaches: the explanatory strength of mathematical equations must be combined with methods from statistical modelling to successfully describe, predict and understand underlying mechanisms.
This proposal unites statistics and mathematics in a tailored collaboration with surface fluid mechanics and neuroscience to bridge the above top-down and bottom-up paradigms.?Data from tailor-made experiments in theses sciences, which are test beds of highly complex systems, will be used to adapt governing equations including uncertainty. These models will allow for qualitative and quantitative properties to be analysed. It is a main goal of the project to develop methods to estimate, forecast and understand complex phenomena for which the governing laws are not yet known.

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3.2BILL. NOKtotal funding in the programme period598PROJECTShave received funding in the programme period2SOURCEShave financed the programme