P. S. Koutsourelakis

Maximilian Stupp, P. S. Koutsourelakis

Coarse-grained (CG) models provide an effective route to reducing the complexity of molecular simulations (MD), but conventional approaches depend heavily on long all-atom MD trajectories to adequately sample configurational space. This data dependence limits accuracy and generalizability, as unvisited configurations remain excluded from the resulting CG models. We introduce a fully data-free, generative framework for CG that directly targets the all-atom Boltzmann distribution. The model defines a structured latent space comprising slow collective variables, associated with multimodal marginal densities capturing metastable states, and fast variables, represented through simple, unimodal conditional distributions. A learnable, bijective map from latent space to atomistic coordinates enables the automatic and accurate reconstruction of molecular structures. Training relies solely on the interatomic potential and minimizes the reverse Kullback-Leibler (KL) divergence via an energy-based objective. To stabilize optimization and ensure mode coverage, we employ an adaptive tempering scheme that promotes the exploration of diverse configurations. Once trained, the model can generate independent, one-shot equilibrium samples at full atomic resolution. Validation on two synthetic systems, a double-well potential and a Gaussian mixture model, as well as on the benchmark alanine dipeptide, demonstrates that the method captures all relevant modes of the Boltzmann distribution, reconstructs atomic configurations, and automatically learns physically meaningful CG representations. These results suggest a promising, data-free alternative to traditional CG techniques, offering both a principled approach to addressing the long-standing "chicken-and-egg" challenge in coarse-graining and an effective solution to the back-mapping problem by enabling accurate reconstruction of all-atom configurations.

Vincent C. Scholz, P. S. Koutsourelakis

We present a novel extension of the Weak Neural Variational Inference (WNVI) framework for probabilistic material property estimation that explicitly quantifies model errors in PDE-based inverse problems. Traditional approaches assume the correctness of all governing equations, including potentially unreliable constitutive laws, which can lead to biased estimates and misinterpretations. Our proposed framework addresses this limitation by distinguishing between reliable governing equations, such as conservation laws, and uncertain constitutive relationships. By treating all state variables as latent random variables, we enforce these equations through separate sets of residuals, leveraging a virtual likelihood approach with weighted residuals. This formulation not only identifies regions where constitutive laws break down but also improves robustness against model uncertainties without relying on a fully trustworthy forward model. We demonstrate the effectiveness of our approach in the context of elastography, showing that it provides a structured, interpretable, and computationally efficient alternative to traditional model error correction techniques. Our findings suggest that the proposed framework enhances the accuracy and reliability of material property estimation by offering a principled way to incorporate uncertainty in constitutive modeling.

L. Felsberger, P. S. Koutsourelakis

The combination of high-dimensionality and disparity of time scales encountered in many problems in computational physics has motivated the development of coarse-grained (CG) models. In this paper, we advocate the paradigm of data-driven discovery for extract- ing governing equations by employing fine-scale simulation data. In particular, we cast the coarse-graining process under a probabilistic state-space model where the transition law dic- tates the evolution of the CG state variables and the emission law the coarse-to-fine map. The directed probabilistic graphical model implied, suggests that given values for the fine- grained (FG) variables, probabilistic inference tools must be employed to identify the cor- responding values for the CG states and to that end, we employ Stochastic Variational In- ference. We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the pre- dictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system. We demonstrate the efficacy of the proposed frame- work in high-dimensional systems of random walkers.

Isabell M. Franck, P. S. Koutsourelakis

This paper is concerned with the numerical solution of model-based, Bayesian inverse problems. We are particularly interested in cases where the cost of each likelihood evaluation (forward-model call) is expensive and the number of un- known (latent) variables is high. This is the setting in many problems in com- putational physics where forward models with nonlinear PDEs are used and the parameters to be calibrated involve spatio-temporarily varying coefficients, which upon discretization give rise to a high-dimensional vector of unknowns. One of the consequences of the well-documented ill-posedness of inverse prob- lems is the possibility of multiple solutions. While such information is contained in the posterior density in Bayesian formulations, the discovery of a single mode, let alone multiple, is a formidable task. The goal of the present paper is two- fold. On one hand, we propose approximate, adaptive inference strategies using mixture densities to capture multi-modal posteriors, and on the other, to ex- tend our work in [1] with regards to effective dimensionality reduction techniques that reveal low-dimensional subspaces where the posterior variance is mostly concentrated. We validate the model proposed by employing Importance Sam- pling which confirms that the bias introduced is small and can be efficiently corrected if the analyst wishes to do so. We demonstrate the performance of the proposed strategy in nonlinear elastography where the identification of the mechanical properties of biological materials can inform non-invasive, medical di- agnosis. The discovery of multiple modes (solutions) in such problems is critical in achieving the diagnostic objectives.

Isabell M. Franck, P. S. Koutsourelakis

This paper presents an efficient Bayesian framework for solving nonlinear, high-dimensional model calibration problems. It is based on a Variational Bayesian formulation that aims at approximating the exact posterior by means of solving an optimization problem over an appropriately selected family of distributions. The goal is two-fold. Firstly, to find lower-dimensional representations of the unknown parameter vector that capture as much as possible of the associated posterior density, and secondly to enable the computation of the approximate posterior density with as few forward calls as possible. We discuss how these objectives can be achieved by using a fully Bayesian argumentation and employing the marginal likelihood or evidence as the ultimate model validation metric for any proposed dimensionality reduction. We demonstrate the performance of the proposed methodology for problems in nonlinear elastography where the identification of the mechanical properties of biological materials can inform non-invasive, medical diagnosis. An Importance Sampling scheme is finally employed in order to validate the results and assess the efficacy of the approximations provided.