# IMod – Project information

# IMod – Partial differential equations, statistics and data:

An interdisciplinary approach to data-based modelling

## Scientific description

IMod is a six-year large-interdisciplinary research project co-funded by the *Research Council of Norway* through their FRIPRO programme and the *Norwegian University of Science and Technology*. Its primary objective is to develop novel mathematical-statistical frameworks for data-driven models of complex systems, guided by problems in fluid mechanics and neuroscience. It is divided into three work packages and seven subprojects, each with a junior researcher position employed by the project. Equally many principal investigators are involved, as well as external partners and affiliated junior researchers contributing to the project.

The **four main subgoals** of the project will be achieved through these subprojects. We will:

**Combine**partial differential equations and statistical theory to develop general models with uncertainty for capturing interactions in complex systems.**Create**effective and fast methods to identify physical parameters from sparsely observed phenomena.- Rigorously
**study**the systems from a mathematical and physical viewpoint. **Unite**theory and data to develop new models in fluid mechanics and neuroscience using tailor-made experiments.

The approximate time plan and activities of the project are illustrated in the following Gantt chart.

The three work packages are described below.

### WP1. Building from data

This package is connected to subgoals 1–3 above.

Its first sub-package centers on building **Partial differential models from generic data**. Time series \(u(t)\) will be captured and described through partial differential equations \( \partial_t u + Lu + N(u) = 0\) where \(L\) is a linear Fourier multiplier operator, and \(N(u)\) is a nonlinear operator, typically realised through a nonlocal pseudo-product. The signal \(u(t)\) is given, and the (parameterised) operators \(L\) and \(N\) are sought as a classification of the signal, or type of signals. The sub-package has one postdoc position directly hired on the project, and is led by Ehrnstrom and Steinsland.

The second sub-package concerns **Coupling partial differential equations to statistical modelling**. The goal of this sub-package is to develop intrinsic couplings between stochastic partial differential equations, on the one hand, and Gaussian random fields, on the other. A starting point is the connection between covariance Matérn functions used in statistical modelling, and the the class of stochastic, fractional and elliptic partial differential equations \( \tau (\kappa^2 - \Delta)^{\alpha} u(s) = W(s)\) in space. One goal is the extension of this framework to time-dependent settings, another is improved error estimates in the stochastic boundary value problem used when the above stochastic partial differential equation is solved in space. The sub-package has one PhD hired directly on the project, and is led by Fuglstad and Jakobsen.

The third sub-package deals with **Uncertainty in nonlinear and nonlocal models**. The sub-package focuses mainly on classical PDEs augmented with noise in an appropriate part, and on systems studied and obtained in a process of statistical averaging over many measurements, data points, or agents (such as particles, neurons or nodes). A nonlinear equation driven by noise is the stochastic Camassa–Holm equation \[0 = du + (u \partial_x u + \partial_x P) dt + (\sigma \partial_x u + (1-\partial_x^2)^{-1} (2 \partial_x \sigma u + \partial_x^2 \sigma \partial_x u)) \circ dW.\] A statistically averaged system is the mean field games equations \[\begin{align*}-\partial_t u + |Du|^2 - Lu &= 0,\\ \partial_t m + \mathrm{div}(mDu) - Lm &= 0.\end{align*}\] The sub-package has one postdoc position directly hired on the project, and is led by Holden, Jakobsen and Ehrnstrom.

### WP2. Mathematical-statistical models in neuroscience

This package unites the theory from subgoals 1–3 to models in neuroscience in subgoal 4.

Its first subpackage is **Development of existing frameworks for forward and top-down modelling**. It deals with existing mathematical neuro-science models such as Hudgin–Huxley, Fokker–Planck and Wilson–Cowan equations, as well as existing models for populations of agents (mean-field games) that have not yet been used in the neuro-science setting. Questions of investigation are stability of neural representations under perturbation of noise, and dynamics of representations through different models. The sub-package has one PhD position directly hired on the project, and is led by Dunn, Jakobsen and Holden.

The second sub-package concerns **Novel data-driven models of temporal–spatial patterns and dynamics**. Its aim is to take one step beyond current state space representations of the brain – which connect features such as neuronal firing-rate to head direction or positions – to models that can possible discover underlying processes and salient features, such as decision making, communication patterns and dynamical fingerprints. The latter exist over time, and would give a dynamical representation of physiological or psychological states or processes. This sub-package also deals with methods for characterising state space representations where the data samples from a mixture of representations; here Bayesian model selection will be used to de-mix populations. The sub-package has one postdoc position directly hired on the project, and is led by Dunn and Ehrnstrom.

### WP3. Surface fluid dynamics as a key to sparse modelling

This package unites the theory from subgoals 1–3 to models in fluid mechanics in subgoal 4.

Its first subpackage is **Turbulence statistics and the study of scales**. The background is the flow of water governed by the Euler and Navier–Stokes equations, in their simplest form written \[\partial_t u + (u\cdot \nabla u) = -\nabla p + F + \nu \nabla^2 u\] Considering turbulence in these equations one may average over slow- and fast-varying scales, yielding the Reynolds-averaged Navier-Stokes equations \[\bar u_j \partial_j \bar u_i = F_i + \partial_j [-p \delta_{ij} + \nu(\partial_j \bar u_l + \partial_i \bar u_j) - \overline{u_l’u_j’}].\] The basic concept is now the covariance matrix \(\mathrm{Cov}(u_i’,u_j’) = \overline{u_l’ u_j’}\). A main aim is to identify variables in the interaction between turbulance and surface waves, that can both be measured experimentally and are amenable to statistical analysis, yielding better and robust models. The sub-package has one PhD position directly hired on the project, and is led by Ellingsen, Fuglstad and Steinsland.

The second subpackage concerns **Dispersive model equations from the lab**. It builds on the theory of WP1 and the experimental setting of the fluid mechanics lab to build, recapture and improve dispersive model equations directly from physical data. The first step is to regain the known linear dispersion relations for water waves through the mathematical–statistical framework in WP1a; in the second step, we extend the framework to include nonlinear interaction. Finally, we wish to study the interaction between surface and interior to capture models that couple the two using only sparse surface information. The sub-package has one postdoc position directly hired on the project, and is led by Ellingsen and Ehrnstrom.

WP3 makes use of two different laboratories, and this is the larger of them.

### Upcoming activities

### Publications

Chowdhury, I., and Jakobsen, E.R. (2024) Precise Error Bounds for Numerical Approximations of Fractional HJB Equations. *IMA J. Numer. Anal.*. DOI: 10.1093/imanum/drae030.

Chowdhury, I., Jakobsen, E.R., and Krupski, M. (2024) On fully nonlinear parabolic mean field games with examples of nonlocal and local diffusions. *SIAM J. Math. Anal.*, **56**(5):6302–6336.

Alibaud, N., Endal, J., and Jakobsen, E.R. (2024) Optimal stability results and nonlinear duality for L-infinity entropy and L1 viscosity solutions. *J. Math. Pures Appl.*, **188**:26–72.

Altay, U., Paige, J., Riebler, A., and Fuglstad, G.-A. (2024) GeoAdjust: Adjusting for Positional Uncertainty in Geostatistial Analysis of DHS Data. *The R journal*.

Weichert, S., Smeltzer, B. K., and Ellingsen, S. Å. (2023) Biases from spectral leakage in remote sensing of near-surface currents. *IEEE Transactions on Geoscience and Remote Sensing*.

Altay, U., Paige, J., Riebler, A., and Fuglstad, G.-A. (2022) Jittering Impacts Raster- and Distance-based Geostatistical Analyses of DHS Data. *Statistical Modelling*. In press.

del Teso, F., Endal, J., Jakobsen, E.R., and Vazquez, J.L. (2023) Evolution Driven by the Infinity Fractional Laplacian. *Calc. Var.*, **62**(136).

del Teso, F., Endal, J., and Jakobsen, E.R. (2023) Uniform tail estimates and Lp-convergence for finite-difference approximations of nonlinear diffusion equations. *Discrete Contin. Dyn. Syst.*, **43**(3&4): 1319–1346.

Zhang, J., Bonas, M., Bolster, D., Fuglstad, G.-A., and Castruccio, S. (2023) High Resolution Global Precipitation Downscaling with Latent Gaussian Models and Nonstationary SPDE Structure. *Journal of the Royal Statistical Society: Series C*. In press.

Lenain, Luc; Smeltzer, Benjamin Keeler; Pizzo, Nick; Maria, Freilich; Colosi, Luke; Ellingsen, Simen Andreas Ådnøy; Grare, Laurent; Peyriere, Hugo; Statom, Nick. (2023) Airborne Remote Sensing of Upper-Ocean and Surface Properties, Currents and Their Gradients From Meso to Submesoscales. *Geophysical Research Letters*, **50**.

Pizzo, Nick; Lenain, Luc; Rømcke, Olav; Ellingsen, Simen Andreas Ådnøy; Smeltzer, Benjamin Keeler. (2023) The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves. *Journal of Fluid Mechanics*, **954**.

Smeltzer, Benjamin Keeler; Rømcke, Olav; Hearst, R. Jason; Ellingsen, Simen Andreas Ådnøy. (2023) Experimental study of the mutual interactions between waves and tailored turbulence. *Journal of Fluid Mechanics*, **962**.

Zheng, Zibo; Li, Yan; Ellingsen, Simen Andreas Ådnøy. (2023) Statistics of weakly nonlinear waves on currents with strong vertical shear. *Physical Review Fluids*, **8**.

Ehrnstrom, Mats; Nik, Katerina; Walker, Christoph. (2022). A direct construction of a full family of Whitham solitary waves. *Proceedings of the American Mathematical Society*, **151**(2).

Ehrnstrom, Mats; Nik, Katerina; Walker, Christoph. (2022). A direct construction of a full family of Whitham solitary waves. *Proceedings of the American Mathematical Society*, **151**(2).

Ehrnstrom, Mats; Nilsson, Dag; Groves, Mark D. (2022) Existence of Davey–Stewartson type solitary waves for the fully dispersive Kadomtsev–Petviashvilii equation. *SIAM Journal on Mathematical Analysis*, **54**(4).

Ehrnstrom, Mats; Wang, Yuexun. (2022) Enhanced existence time of solutions to evolution equations of Whitham type. *Discrete and Continuous Dynamical Systems. Series A, ***42**(8).

Paige, J., Fuglstad, G.-A., Riebler, A., and Wakefield, J. (2022). Spatial Aggregation with Respect to a Population Distribution: Impact on Inference. *Spatial Statistics*, **52.**

### Preprints

Altay, U., Paige, J., Riebler, A., and Fuglstad, G.-A.. (2023) GeoAdjust: Adjusting for Positional Uncertainty in Geostatistial Analysis of DHS Data. arXiv:2303.12668.