Orestis Oikonomou

Orestis Oikonomou, Levi Lingsch, Dana Grund et al.

Analytical solutions of differential equations offer exact insights into fundamental behaviors of physical processes. Their application, however, is limited as finding these solutions is difficult. To overcome this limitation, we combine two key insights. First, constructing an analytical solution requires a composition of foundational solution components. Second, iterative solvers define parameterized function spaces with constraint-based updates. Our approach merges compositional differential equation solution techniques with iterative refinement by using formal grammars, building a rich space of candidate solutions that are embedded into a low-dimensional (continuous) latent manifold for probabilistic exploration. This integration unifies numerical and symbolic differential equation solvers via a neuro-symbolic AI framework to find analytical solutions of a wide variety of differential equations. By systematically constructing candidate expressions and applying constraint-based refinement, we overcome longstanding barriers to extract such closed-form solutions. We illustrate advantages over commercial solvers, symbolic methods, and approximate neural networks on a diverse set of problems, demonstrating both generality and accuracy.

Hyeongkyun Kim, Orestis Oikonomou

Flood susceptibility mapping (FSM) is vital for disaster prevention but remains challenging in data-scarce regions where hydrodynamic models require dense geophysical inputs. This work introduces ZeroFlood, a geospatial foundation model framework for data-efficient FSM. The approach fine-tunes Geospatial Foundation Models (GFMs) with Thinking-in-Modality (TiM) reasoning, enabling flood prediction from basic Earth observation data such as Sentinel-1 or Sentinel-2 imagery. Using paired EO and simulated flood maps from data-rich regions, ZeroFlood bridges data availability gaps through cross-modal representation learning. Experiments with TerraMind and Prithvi GFMs show that TiM enhances model robustness, with the TerraMind-Large configuration achieving an F1 score of 67.21. The results demonstrate the feasibility of foundation-model-based FSM as a scalable and data-efficient solution for flood risk management.