Lapland Scientific Retreat 2026

The accurate reconstruction and prediction of the near-Earth space environment are crucial for anomaly detection, empirical model development, and a deeper understanding of physical processes. Data assimilation (DA) provides a powerful framework for obtaining reanalysis and forecasts that integrate model and observational uncertainties. This work reviews recent advances in the application of Kalman filtering tech-niques to the outer radiation belt system. We discuss the assimilation of electron flux observations into diffusion models of the radiation belts, followed by the implementation of a split-operator technique to address challenges encountered when applying full 3D Kalman filters. Additionally, we highlight the development of 3D split-operator standard Kalman filter, together with extensions to higher-dimensional ensemble Kalman Filtering approaches. The application of Kalman filtering techniques has been demonstrated in the development of a real-time forecasting of radiation belt electron fluxes. Furthermore, in this presentation we explore the uses of DA-based reanalysis for not just global state analysis of the radiation belts, but also for potential identification of missing physical processes, error assessment using innovation vectors, and intercalibration of data assimilation techniques across multiple missions.

We discuss prior fields given in the form of spatially inhomogeneous SPDE solutions in the context of functional Bayesian inverse problems and uncertainty quantification. Both theoretical and numerical results are considered. Joint work with Babak Maboudi Afkham, Mirza Karamehmedovic, Lassi Roininen.

Integrative factor models have proven to be crucial for identifying reproducible biological pathways shared by different cancer studies that traditional factor analysis approaches may miss due to systematic biases. Existing integrative factor models, while valuable, often neglect the impact of covariates and confounders, introducing bias into the signal and/or lack a study-specific factor structure totally independent of the common latent structure. Moreover, these models require post-processing steps for loadings, such as varimax rotation, and restrictions on the loading matrix for identifiability, essential for interpretation. To address these challenges, we present a novel class of integrative factor models: "Rotational Invariant Sparse Factor Models" (RISFM). RISFM displays several advantages: 1) Providing sparse low-dimensional common and study-specific factors while adjusting for confounding effects using Bayesian methods. 2) Addressing identifiability issues crucial for interpretation through the l1-ball prior. 3) Ensuring computational efficiency for practical applicability. We validate the proposed RISFM approach through extensive simulations and its application to hepatocellular carcinoma human-mice gene expression cancer data. The results showcase the utility of RISFM to determine the overall human-mice genomic similarities, to identify the most appropriate mouse model for studying different human patient sub-populations, and to obtain the co-regulation mechanisms that are idiosyncratic to only humans or mice. The main goal is to improve the current understanding of mouse mutation models and contribute to develop new ones for precision medicine.

Bayesian filtering is a fundamental methodology for online inference in stochastic dynamical systems, but the performance of practical algorithms deteriorates severely in high-dimensional settings, where standard methods often become unstable or inaccurate. In this work we investigate a class of constrained Bayesian filtering methods designed to address this challenge in discretely-observed diffusion systems. The key idea is to restrict the filtering distribution to suitably chosen feasible sets that reflect structural properties of the model and the data, thereby excluding unstable or implausible states while retaining the essential statistical information. We develop a theoretical framework to analyse the stability and accuracy of such constrained filters, establishing conditions under which they provide convergent approximations with explicit rates. Finally, we discuss practical implementation issues and present numerical experiments which show how constrained Bayesian filtering offers a robust and scalable approach to online inference.

Tailoring medical treatment strategies to the covariates of specific patient subgroups or even to individuals has shown highly promising results in improving therapeutic outcomes. To enable personalized therapeutic approaches, a first essential step is to identify or learn the causal relationships between medical interventions, patient covariates and disease progression or improvement. We illustrate this using a real-world data challenge on cellular information retrieval. Machine-learning methods such as Random Forests can provide deeper insights into potentially relevant biomarkers that distinguish healthy from diseased groups. In the context of rare immune disorders, such methods allow hidden relationships to be uncovered in a significantly shorter time frame compared to manual analysis. Because the data is usually collected across multiple days, transport mapping is required to appropriately normalize the distributions to ensure technical variation does not interfere with patient-specific individuality. We further investigate under which mathematical conditions this transformation is justified. Once these relationships are established and a predictive model has been constructed, sequentially incoming patient data can be integrated with model predictions. This enables the use of rein- forcement-learning techniques for optimal decision-making in the presence of uncertainty.

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Bayesian inference for models with intractable likelihoods requires balancing accuracy and computational cost. We propose an amortized MCMC-based approach that matches the accuracy of the exchange algorithm while significantly reducing computation. A gradient-informed grid selection combined with Hermite interpolation yields an accurate and efficient surrogate model, as demonstrated by application to a Potts model.

We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution for uncertainty quantification (UQ), and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields, and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyse in abstract form neural fields that have been spatially discretised, setting the foundations for analysing UQ schemes.

Satellite observations play a critical role in numerical weather prediction. In the traditional Ensemble Kalman Filter, these observations are assimilated by weighting their associated errors against model uncertainties to produce an optimal estimate. This process requires radiative transfer simulations for passive, downward-viewing satellite radiometers operating in the visible, infrared, and microwave spectra. Typically, such simulations rely on numerically integrating physical laws via models like RTTOV. In talk, we discuss two data-driven surrogate operators: First, a fully data driven surrogate operator and second, a data driven correction operator for radiative transfer based on modern machine-learning architectures from computer vision. Whereas the former is a fully data-driven emulator, for the latter, our method adopts an incremental, hybrid formulation: the network learns only the residuals with respect to RTTOV, thereby embedding established radiative-transfer physics into the surrogate while enabling data-driven refinement in complex, cloud-affected conditions.

Neural Graph Pattern Machines (GPMs) have established themselves as a powerful paradigm, empirically outperforming Message Passing Neural Networks (MPNNs) by overcoming the 1-Weisfeiler-Leman expressivity upper bound. However, while their ability to distinguish more graphs than 1-WL is understood, a rigorous framework explaining their statistical stability and generalization capabilities on molecular tasks remains missing. In this work, we bridge this gap by grounding GPMs in the theory of Graph Limits. We demonstrate that stochastic pattern sampling is not merely a heuristic for feature extraction, but an efficient Monte-Carlo estimator for Homomorphism Densities, which form a complete basis for the topology of sparse graphs, the ”Graphings”. This perspective shifts the focus from discrete separation to continuous estimability: We utilize the Benjamini-Schramm topology to formally characterize which molecular properties are learnable independent of system size, and which are not. Building on this, we address critical shortcomings of current path-based GPMs: (i) We introduce a Multi-Pattern Basis to capture degree moments and spectral properties that paths miss; (ii) We extend the notion of graphhomomorphisms to the 3D space of molecules and prove their Lipschitz stability to resolve chirality; and (iii) We extend the from attention-based correlations introduced by the GPM to Structural Interventions, enabling causal reasoning and explainability. Our framework provides the theoretical guarantee that GPMs do not just memorize finite structures, but approximate the underlying physical laws of the molecular limit object.

In state and parameter estimation problems, nested particle filters are a particularly suitable approach for sequentially incoming data, as they enable uncertainty quantification while simultaneously approximating the latent signal and the model parameters. However, the computational complexity associated with the large number of required particles often renders such methods impractical. In this work, we propose an adaptive sampling scheme that dynamically adjusts the number of particles at both the inner and outer levels, thereby reducing computational cost while maintaining acceptable levels of estimation accuracy.

We study the application of quasi-Monte Carlo (QMC) methods for Bayesian inverse problems governed by PDEs. For the parameterization of the unknown quantities, we consider a model recently studied by Chernov and Le [1,2] as well as Harbrecht, Schmidlin, and Schwab [3] in which the input random field is assumed to belong to a Gevrey class. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Specifically, we consider the application of QMC for Bayesian shape inversion [4] and electrical impedance tomography [5] using the techniques developed in [6].
References:
[1] A. Chernov and T. Le. Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications. SIAM J. Numer. Anal., 62(4):1874-1900, 2024.
[2] A. Chernov and T. Le. Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification. Comput. Math. Appl., 164:116-130, 2024.
[3] H. Harbrecht, M. Schmidlin, and Ch. Schwab. The Gevrey class implicit mapping theorem with applications to UQ of semilinear elliptic PDEs. Math. Models Methods Appl. Sci., 34(5):881-917, 2024.
[4] A. Djurdjevac, V. Kaarnioja, M. Orteu, and C. Schillings. Quasi-Monte Carlo for Bayesian shape inversion governed by the Poisson problem subject to Gevrey regular domain deformations. To appear in Monte Carlo and Quasi-Monte Carlo Methods 2024, B. Feng and C. Lemieux (eds.), Springer Verlag, 2026.
[5] L. Bazahica, V. Kaarnioja, and L. Roininen. Uncertainty quantification for electrical impedance tomography using quasi-Monte Carlo methods. Inverse Problems 41, 065002, 2025.
[6] V. Kaarnioja and C. Schillings. Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs. Preprint 2024, arXiv:2405.03529 [math.NA].

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I will introduce our new Research Council of Finland project titled "Enhanced Wood Tracing Systems for Sustainable Forestry and Improved Wood Utilization". This is a joint project between LUT University, University of Eastern Finland, Finnish Geospatial Research Institute, and University of Helsinki.