arXiv subjects

Leonhard Hennicke

Publications and source records attributed to Leonhard Hennicke.

3 recordsLinked to original sources

arXiv2026

Efficient Message Passing for Partial Differential Equation Priors

Prior information for real-world physical quantities is most elegantly expressed via partial differential equations (PDEs). In this paper, we propose a novel way to solve PDEs using probabilistic inference on a factor graph. In general, factor graphs provide a natural way to encode prior knowledge into a model as explicit factors; here, this knowledge is provided by a governing PDE, which narrows the solution space, while observed data further shape the posterior over the parameters. The approximate parameter posterior is inferred using message passing based on moment matching, without posterior sampling or global gradient-based optimization. We demonstrate our approach on the first-order advection and the second-order semi-linear Fisher-KPP equations, where it achieves predictive accuracy comparable to a standard baseline while providing structured predictive uncertainty. Moreover, the inferred posterior marginal means and uncertainty structure match more closely those obtained using Hamiltonian Monte Carlo than the evaluated mean-field variational inference baseline, while requiring up to 10x less training time in our experiments, with inference speed comparable to variational inference.

cs.LG↗

arXiv2026

Direct Message Approximation (DMA): A Consistency-Based Framework for Tractable Approximate Inference on Factor Graphs

Approximate message passing on factor graphs underlies two dominant families of probabilistic inference algorithms: expectation propagation (EP) and variational message passing (VMP). Both methods approximate the marginal at each factor edge, forcing an iterative round-robin schedule, risking negative-precision messages, and, for VMP, collapsing to point estimates at Dirac-delta factors. We introduce Direct Message Approximation (DMA), which approximates factor-to-variable messages directly rather than the marginal. For normalisable factors, we define a consistency condition (requiring exactness when all other incoming messages are Dirac deltas) to guide message construction. We prove a master theorem (proper messages, any graph) bounding marginal KL from message KL, with three structural corollaries: Dirac-input consistency, no EP-style inner-loop iteration, and no negative-precision messages. Further, we prove a complementary $O(1/r^2)$ guarantee for the inherently improper backward message of the product factor, whose closed-form treatment has resisted prior work. As a concrete instantiation, we derive explicit DMA messages for the product and leaky-ReLU factors and assemble a Bayesian neural network (BNN) inference algorithm with one forward/backward sweep per training example and no gradient learning-rate hyperparameter, validating that the structural guarantees translate to predictive uncertainty that widens in data-sparse regions, including under model mismatch.

cs.LG↗

arXiv2024

Mind the Gap Between Synthetic and Real: Utilizing Transfer Learning to Probe the Boundaries of Stable Diffusion Generated Data

Generative foundation models like Stable Diffusion comprise a diverse spectrum of knowledge in computer vision with the potential for transfer learning, e.g., via generating data to train student models for downstream tasks. This could circumvent the necessity of collecting labeled real-world data, thereby presenting a form of data-free knowledge distillation. However, the resultant student models show a significant drop in accuracy compared to models trained on real data. We investigate possible causes for this drop and focus on the role of the different layers of the student model. By training these layers using either real or synthetic data, we reveal that the drop mainly stems from the model's final layers. Further, we briefly investigate other factors, such as differences in data-normalization between synthetic and real, the impact of data augmentations, texture vs.\ shape learning, and assuming oracle prompts. While we find that some of those factors can have an impact, they are not sufficient to close the gap towards real data. Building upon our insights that mainly later layers are responsible for the drop, we investigate the data-efficiency of fine-tuning a synthetically trained model with real data applied to only those last layers. Our results suggest an improved trade-off between the amount of real training data used and the model's accuracy. Our findings contribute to the understanding of the gap between synthetic and real data and indicate solutions to mitigate the scarcity of labeled real data.

cs.CV↗