An Adaptive-Horizon Graph Neural Operator for coupled surrogate morphodynamic modelling

Operational flood forecasting still relies on high-fidelity two-dimensional hydraulic solvers, but their runtime can be prohibitive for rapid decision support on large urban floodplains. In parallel, AI-based surrogate models have shown strong potential in several areas of computational physics for accelerating otherwise expensive high-fidelity simulations. We address this issue on the lower Têt River (France), starting from a production-grade Telemac2D model defined on a high-resolution unstructured finite-element mesh with more than $4\times 10^5$ nodes. From this setup, we build a learning-ready database of synthetic but operationally grounded flood events covering several representative hydrograph families and peak discharges. On top of this database, we develop a graph-neural surrogate based on projected meshes and multimesh connectivity. The projected-mesh strategy keeps training tractable while preserving high-fidelity supervision from the original Telemac simulations, and the multimesh construction enlarges the effective spatial receptive field without increasing network depth. We further study the effect of an explicit discharge feature $Q(t)$ and of pushforward training for long autoregressive rollouts. The experiments show that conditioning on $Q(t)$ is essential in this boundary-driven setting, that multimesh connectivity brings additional gains once the model is properly conditioned, and that pushforward further improves rollout stability. Among the tested configurations, the combination of $Q(t)$, multimesh connectivity, and pushforward provides the best overall results. These gains are observed both on hydraulic variables over the surrogate mesh and on inundation maps interpolated onto a common $25\,\mathrm{m}$ regular grid and compared against the original high-resolution Telemac solution. On the studied case, the learned surrogate produces 6-hour predictions in about $0.4\,\mathrm{s}$ on a single NVIDIA A100 GPU, compared with about $180\,\mathrm{min}$ on 56 CPU cores for the reference simulation. These results support graph-based surrogates as practical complements to industrial hydraulic solvers for operational flood mapping.

Spatiotemporal forecasting on hydrologic graphs is especially prone to instability in open-boundary systems, where the forecast domain exchanges fluxes with an unobserved exterior. In such systems, boundary nodes receive external forcing, e.g., upstream inflows in rivers or tidal signals in coastal regions, that is typically unavailable at prediction time. The absence of this information can compound errors as forecasts unfold in an autoregressive fashion, leading to inferior long-horizon performance. This paper dissects this instability issue by exploring two questions. 1) What boundary forcing enters the forecast domain when information beyond the boundary is missing? 2) How should this forcing propagate through the domain without incurring error amplification under autoregressive rollout? To address both, we propose a new computing framework comprising two key components. First, to compensate for the boundary forcing, our framework learns ghost node proxies from the boundary and interior nodes, striving to approximate unobserved external inputs. Second, to control error accumulation from these learned proxies, we leverage two physics refiners. In particular, one refiner enforces local consistency by aligning ghost proxies with their two-hop neighbors (i.e., boundary nodes and their immediate interiors). The other refiner enhances global stability by correcting the model forecasts through a physics-guided graph neural operator, reducing long-horizon numerical drift. Two real-world hydrologic graphs are employed for empirical evaluation. Comparative results show that our proposal enjoys higher prediction accuracy and long-horizon stability over both learning-based and physics-informed model competitors.

Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical states. Learning these maps is difficult, especially for time-dependent systems that must assimilate history and remain stable under autoregressive rollout. Many neural operators work best on regular, structured grids, while realistic simulations often require unstructured meshes or point clouds to resolve complex geometries; in such settings, grid-centric representations can lose accuracy. Graph neural operators handle these domains through message passing or spectral graph filtering, but pairwise edges do not directly capture group-wise couplings among mesh cells, local neighborhoods, or conservation volumes. We introduce the Hypergraph Adaptive waveLet Operator (HALO), which lifts the domain to a hypergraph and learns in its spectral wavelet domain. HALO avoids explicit hypergraph-Laplacian eigendecomposition through Chebyshev polynomial wavelet filters, giving localized spectral kernels at linear sparse-matrix cost. Its trainable dyadic wavelet scales are regularized toward tight-frame coverage, allowing the frequency response to adapt to each PDE while encouraging stable multi-scale spectral coverage. Across 2D and 3D benchmarks on structured and unstructured discretizations, HALO achieves best or near-best accuracy among frequency-, transformer-, DeepONet-, state-space-, and graph-based baselines and sustains stable multi-step rollouts. The same model scales to industrial aerodynamic geometries: on meshes of a few hundred thousand points it is on par with, or better than, the strongest fixed-discretization transformers, while remaining resolution-equivariant.