Special Session 113: Recent Advances in Uncertainty Quantification and Scientific Machine Learning with Applications to Complex Dynamical Systems

Assimilative Causal Inference: Tracing Causes from Effects to Predict and Attribute Significant Events

Marios Andreou University of Wisconsin-Madison USA Co-Author(s):    Nan Chen, Erik Bollt

Abstract: Causal inference is fundamental across scientific disciplines, yet state-of-the-art methods often struggle to capture instantaneous causal relationships in high-dimensional systems. This work introduces assimilative causal inference (ACI), a paradigm-shifting framework that reframes causality as a Bayesian inverse problem using data assimilation. Rather than measuring forward influence from causes to effects, ACI instead traces causality backwards by quantifying how incorporating future information from effects reduces the uncertainty in the estimated system state. As such, effects are interpolated onto causes, contrasting classical predictive approaches that extrapolate causes forward to identify effects. ACI dynamically determines causal interactions without observing candidate causes, accommodates short datasets, and scales efficiently to high dimensions. Crucially, it provides online tracking of causal roles, which may reverse intermittently, and facilitates rigorous criteria for the causal influence range (CIR) of a relationship. The ACI-based CIR metric is objectively defined, without empirical thresholds, and admits both forward- and backward-in-time formulations. The forward CIR quantifies the temporal reach of a cause, while the backward CIR traces the onset of triggers for an observed effect, enabling causal predictability and attribution in transient regimes. The effectiveness of ACI and its CIR framework is demonstrated on nonlinear dynamical systems showcasing intermittency, extreme events, and tipping points.

New results in feedback particle filter

Sumith Reddy Anugu TU Ilmenau Germany Co-Author(s):    Sumith Reddy Anugu (TU Ilmenau), Jana de Wiljes (TU Ilmenau), Gottfried Hastermann (TU Ilmenau)

Abstract: Feedback particle filter is known to provide a way to compute the filter (conditional) distribution of the associated filtering model. In this talk, we present certain new results in the context of feedback particle filter. We first present the well-posedness of the associated Poisson equation (which is crucial to the implementation of the feedback particle filter) under certain easily verifiable sufficient conditions. Then, we introduce a new approximation procedure for the feedback particle filter (that is computationally more efficient), which makes use of a `local` version of the associated Poisson equation.

Learning data assimilation from artificial intelligence

Marc BOCQUET \`Ecole nationale des ponts et chauss\`ees France Co-Author(s):    Marc Bocquet and collaborators

Abstract: We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning. The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with low-order dynamics which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter, and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system. This also suggests building a new class of efficient deep learning-based ensemble-free DA algorithms.

Multiscale Modelling and Data Assimilation for Sea Ice Dynamics

Quanling Deng Yau Mathematical Sciences Center Peoples Rep of China Co-Author(s):

Abstract: Sea ice dynamics involve complex interactions across a wide range of scales, from individual floe collisions to large-scale coupled behaviour with the ocean and atmosphere. This talk presents a multiscale perspective on sea ice modelling and data assimilation, with an emphasis on connecting discrete floe-level dynamics to continuum descriptions suitable for larger-scale prediction and the corresponding sea ice rheology. After a brief introduction to the main features of sea ice and a short overview of major continuum approaches, I will introduce a multiscale modelling framework for sea ice floes and discuss how it captures essential physical mechanisms across scales. Building on this framework, I will then present a multiscale data assimilation approach designed to combine Lagrangian and Eulerian observational data in a coherent and flexible way. The talk will conclude with a brief discussion of ongoing efforts to incorporate machine learning tools into the assimilation pipeline, with the goal of improving predictive accuracy, robustness, and computational efficiency in sea ice modelling.

Spatial Jump Process Models for Estimating Antibody-Antigen Interactions

Samuel Isaacson Boston University USA Co-Author(s):

Abstract: Surface Plasmon Resonance (SPR) assays are a standard approach for quantifying kinetic parameters in antibody-antigen binding reactions. Classical SPR approaches ignore the bivalent structure of antibodies, and use simplified ODE models to estimate effective reaction rates for such interactions. In this work we develop a new SPR protocol, coupling a model that explicitly accounts for the bivalent nature of such interactions and the limited spatial distance over which such interactions can occur, to a SPR assay that provides more features in the generated data. Our approach allows the estimation of bivalent binding kinetics and the spatial extent over which antibodies and antigens can interact, while also providing substantially more robust fits to experimental data compared to classical ODE models. I will present our new modeling and parameter estimation approach, and demonstrate how it is being used to study interactions between antibodies and spike protein. I will also explain how we make the overall parameter estimation problem computationally feasible via the construction of a surrogate approximation to the (computationally-expensive) particle model. The latter enables fitting of model parameters via standard optimization approaches.

Capturing Prediction Uncertainty in Data Assimilation

Tijana Janjic KU Eichstaett Ingolstadt Germany Co-Author(s):    Catherine George, Alireza Javanmardi, Eyke Huellermeier

Abstract: Quantification of evolving uncertainties is required for both probabilistic forecasting and data assimilation in weather prediction. In current practice, the ensemble of model simulations is often used as primary tool to describe the required uncertainties. In idealized settings, using toy models that mimic convective situations, we explore two alternative approaches, so called stochastic Galerkin method which integrates uncertainties forward in time using a spectral approximation in the stochastic space and machine learning approach based on conformal prediction.

On the Potential and Pitfalls of Flow Matching for Probabilistic Forecasting

Soon Hoe Lim KTH Royal Institute of Technology and Nordita Sweden Co-Author(s):    Shizheng Lin, Michael Mahoney, N. Benjamin Erichson

Abstract: Flow matching has emerged as a powerful paradigm for generative modeling. From the viewpoint of dynamical measure transport, it constructs continuous-time ODE samplers by prescribing probability paths that connect a base and a target distribution. In this talk, we discuss both the potential and pitfalls of flow matching for probabilistic forecasting of dynamical systems. We observe that forecasting performance is highly sensitive to the choice of probability path, motivating principled constructions that lead to improved computational efficiency and predictive performance on spatio-temporal dynamical system benchmarks. We then revisit flow matching in the empirical setting and analyze the structure of the induced velocity fields, uncovering connections between flow matching, memory effects, and nonparametric dynamical systems. This perspective leads to new sampling strategies and raises questions about the role of parameterization in learning dynamical systems from data.

Sequential Monte Carlo for Bayesian Inference Using Randomized Likelihoods

Josef Martinek Heidelberg University Germany Co-Author(s):    Julian Hofstadler, Alexander M.G. Cox, Robert Scheichl

Abstract: In this talk we focus on Bayesian inverse problems in which the forward parameter-to-observable map is approximated in a stochastic way, for instance by Monte Carlo (MC) simulations. Such problems arise for example in uncertainty quantification of particle transport, where the parameter-to-observable map is defined through the solution of the Boltzmann equation. Approximating the likelihood with high accuracy requires MC simulations with many samples, which makes sampling from the posterior distribution expensive. We present an efficient method for sampling from the posterior based on pseudo-marginal sequential Monte Carlo (SMC) using likelihood tempering. To accelerate sampling from the posterior the method adopts a multilevel approach. A significant speedup is achieved compared to a single-level SMC using high-fidelity MC simulations while achieving the same error, which is verified theoretically and demonstrated by numerical experiments.

HClimRep: a Foundation Model for capturing interactions between the atmosphere, ocean, and sea ice

Savvas Melidonis Forschungszentrum Juelich GmbH Germany Co-Author(s):    Savvas Melidonis, Ankit Patnala, Belkis Asma Semcheddine, Jehangir Awan, Simon Grasse, Kacper Nowak, Aleksei Koldunov, Julius Polz, Enxhi Kreshpa, Illaria Luise, Christian Lessig, Nikolay Koldunov, Thomas Jung, Martin Schultz

Abstract: Climate change presents critical challenges to ecosystems and human society. Accurate projections of climate change and its consequences are essential for assessing climate policies and developing proactive strategies to mitigate extreme weather events. While conventional climate models - grounded in fluid dynamics and radiative transfer physics - have yielded valuable insights, they suffer from several shortcomings: inherent biases, coarse spatial resolution, structural errors, and considerable computational inefficiency. Within Helmholtz Foundation Model Initiative (HFMI), we introduce HClimRep, an AI-driven foundation model designed to capture the intricate coupled dynamics of the atmosphere, stratosphere, ocean, and sea-ice and to generate realistic climate simulations. HClimRep builds on the WeatherGenerator prototype - a EU funded HORIZON weather forecasting project - by expanding its architecture and training regime to deliver true climate projection capabilities. In this presentation we will outline the most recent advances of our project in subseasonal to seasonal forecasting and showcase a atmosphere-ocean downscaling downstream application. By harnessing transformer-based modelling, GPU-level optimization, and the latest developments at the intersection of high performance computing and AI, our overarching goal is to deliver a breakthrough climate research tool that deepens our understanding of the Earth system and underpins evidence based climate adaptation and policy decisions.

Dynamical Low-Rank Ensemble Kalman filtering

Fabio Nobile Ecole Polytechnique Federale de Lausanne (EPFL) Switzerland Co-Author(s):    Sebastien Riffaud, Thomas Trigo Trindade

Abstract: We consider a high-dimensional time-continuously partially observed linear system of SDEs and the Ensemble Kalman filter (EnKF) to evolve the filtering distribution. To reduce computational complexity, we propose a dynamical low-rank approximation (DLR-EnKF), where particles evolve in a relatively small, online-computed, time-varying subspace at reduced cost. This allows for a significantly larger ensemble size compared with standard EnKF at equivalent cost, thereby lowering the Monte Carlo error and improving filter accuracy. Some theoretical properties, including a propagation-of-chaos result, will be presented. We then extend the framework to high-dimensional non-linear dynamical systems for efficient joint state-parameter estimation. We discuss, in particular, a time-integration strategy that combines the Basis Update & Galerkin scheme with forecast/analysis discretisation, and a DEIM-based hyper-reduction technique for efficient evaluation of nonlinear terms. We demonstrate the effectiveness, robustness, and computational advantages of the proposed approach on benchmark problems, including a state/parameter estimation problem for a reduced one-dimensional blood flow model of the human arterial system observed in only three spatial locations.

Geometric extremal graphical models

Ioannis IP Papastathopoulos University of Edinburgh Scotland Co-Author(s):    Lambert De Monte, XIndi Song

Abstract: We study statistical models for multivariate extremes based on transport to a center-outward reference distribution. Our approach combines optimal-transport-based and flow-matching ideas to learn distributional structure in the bulk while retaining geometric features of the tail. In particular, transport to a product-uniform reference provides a natural way to encode radial and angular extremal behaviour, and to examine how inverse rays and transport contours reflect tail geometry. We discuss both Brenier-type and entropic regularizations, with emphasis on the trade-off between smoothness, invertibility, and fidelity in the extremes. The resulting perspective points toward a broader connection between multivariate regular variation, geometric extreme-value analysis, and modern generative modeling.We introduce geometric extremal graphical models, a new framework for describing dependence in multivariate extreme events. The approach is based on a geometric representation of the limiting behaviour of suitably scaled random vectors with light-tailed margins. For block graphs, we show how different measures of extremal dependence propagate through the graph. We focus in particular on measures connected to conditional extreme value theory, which are useful when extreme events occur in some variables without requiring all variables to be extreme at the same time. We also discuss the case of joint extreme behaviour, where several variables become extreme together. Together with recent work linking geometric ideas in multivariate extremes to practical statistical models, these results open the way to modelling high-dimensional extremes with complex dependence structures.

Reduced-order models for data assimilation and uncertainty quantification of multiscale turbulent systems

Di Qi Purdue University USA Co-Author(s):    Di Qi and Jian-Guo Liu

Abstract: Reduced-order data assimilation models for predicting probability distributions of multiscale turbulent systems Abstract: A new strategy is presented for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions. The capability of using reduced-order models to capture key statistical features is investigated. A closed stochastic-statistical modeling framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the random batch method. Effective nonlinear ensemble filters are developed based on the nonlinear coupling structures of the explicit stochastic and statistical equations, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures.

Data assimilation with machine-learned dynamics

Daniel Sanz-Alonso University of Chicago USA Co-Author(s):    Nisha Chandramoorthy, Nathan Waniorek, Wenwen Li

Abstract: Scientific machine learning is transforming the field of data assimilation by providing accurate and cheap-to-evaluate surrogate models for many dynamical systems. In the first part of this talk, I will show that Fourier Neural Operators (FNOs) enjoy polynomial sample complexity for estimating the solution map of a wide class of evolution PDEs. Our theory leverages that these PDEs can be accurately approximated via spectral methods, and that these numerical schemes can in turn be efficiently learned using FNOs. In the second part of this talk, I will show that ensemble Kalman filters with machine-learned surrogate models achieve long-time filter accuracy under standard observability assumptions on the true dynamics and observation models.

Global Convergence of Adjoint-Optimized Neural PDEs

Konstantinos Spiliopoulos Boston University USA Co-Author(s):

Abstract: Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from observed data in order to approximate missing or unresolved physics in the PDE model. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to the available ground truth data by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. These neural PDE models have emerged as an important research area in scientific machine learning. In this work, we study the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity. Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove convergence of the trained neural-network PDE solution to the target data (i.e., a global minimizer).  The theoretical results are illustrated and empirically validated by numerical studies.

Analysis of Data-Driven Smoothing and Forecasting

Andrew Stuart Caltech USA Co-Author(s):    Edoardo Calvello, Elizabeth Carlson, Nikola Kovachki, Michael N. Manta, and Andrew M. Stuart

Abstract: Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation, and for forecasting of dynamical systems, the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing and forecasting problems. The theoretical framework relies on two key components (i) establishing the existence of the mapping to be learned (ii) the properties of the machine learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish the first universal approximation theorem for purely data-driven algorithms for data assimilation and forecasting of dynamical systems. We work in continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz 63, Lorenz 96 and Kuramoto-Sivashinsky dynamical systems.

Neural network surrogates with uncertainty quantification for inverse problems

Aretha L Teckentrup University of Edinburgh Scotland Co-Author(s):    Christian Jimenez Beltran, Antonio Vergari, Konstantinos Zygalakis

Abstract: Inverse problems in differential equations are central to many scientific and engineering applications, requiring the estimation of model parameters based on noisy or incomplete observations. Traditional numerical methods for solving these problems are computationally expensive, particularly in Bayesian settings, where likelihood evaluations must be performed repeatedly in high-dimensional parameter spaces. In this talk, we investigate neural networks as surrogates to address these challenges. By incorporating a Laplace Approximation, our method efficiently approximates the forward model and provides uncertainty estimates. Compared to traditional methods, this approach significantly reduces computational costs while maintaining accurate posterior approximations. These findings underscore the potential of neural networks for scalable and reliable solutions to inverse problems in complex systems.