Physics-Refined Spatiotemporal Forecasting on Open-Boundary Hydrologic Graphs

Haoyang Jiang; Zhengui Wang; Shenghan Gao; Y. Joseph Zhang; Xingquan Zhu; Yi He

cs.LGOct 1, 2026

Physics-Refined Spatiotemporal Forecasting on Open-Boundary Hydrologic Graphs

Authors: Haoyang Jiang, Zhengui Wang, Shenghan Gao, Y. Joseph Zhang, Xingquan Zhu, Yi He

Organizations: William & Mary, Williamsburg, VA, USA · Virginia Institute of Marine Science, William & Mary, Gloucester Point, VA, USA · Florida Atlantic University, Boca Raton, FL, USA

Abstract

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.

Figures & tables

Panel of Figure 1 — Fig. 1 : Demonstration of downstream error amplification in the Danube river system [ 2 ] . (a) Boundary nodes regulate the influx of entire network due to unobserved external forcing, and (b) boundary errors propagate and amplify toward downstream under autoregressive rollout in STGNN.

Fig. 2 : Overall pipeline of PRSOBG . For directed graphs (e.g., the Danube river system) and unstructured meshes (e.g., Chesapeake Bay), PRSOBG makes missing open-boundary forcing learnable by constructing ghost-node proxies from boundary–interior dynamics (Sec. III-A ). The proxy is instantiated as one-to-one upstream explicit ghost nodes in directed graphs, and as three mesh-specific ghost variants in unstructured meshes to inject boundary forcing into message passing (Sec. III-B ). During autoregressive rollout, PRSOBG first applies a local boundary refiner to stabilize proxy–boundary consistency (Sec. III-C ), then applies a global physics refiner to regularize propagation, using either an explicit refiner (Sec. III-D1 ) or a convexified semi-implicit refiner (Sec. III-D2 ).

	$T=6$		$T=16$		$T=30$
Method	RMSE	Imp.	RMSE	Imp.	RMSE	Imp.
Data-Driven Baselines
STGNN (Backbone)	0.3499	–	0.4147	–	0.5924	–
GRU (Temporal Only)	0.2623	+25.04%	0.3746	+9.67%	0.5529	+6.67%
Ghost-Only STGNN	0.2777	+20.63%	0.3878	+6.49%	0.4836	+18.37%
GraphNODE [ 18 ]	0.3466	+0.94%	0.4104	+1.04%	0.5871	+0.89%

TABLE I : River System under autoregressive rollout with

T\in\{6,16,30\}

. Imp. is relative to STGNN (Backbone).

	$T=6$		$T=16$		$T=30$
Method	RMSE	Imp.	RMSE	Imp.	RMSE	Imp.
Data-Driven Baselines
STGNN (Backbone)	0.0732	–	0.1797	–	0.2864	–
GRU (Temporal Only)	0.0524	+28.42%	0.1202	+33.11%	0.2108	+26.40%
GraphNODE [ 18 ]	0.0721	+1.50%	0.1743	+3.01%	0.2782	+2.86%
GraphSSM [ 19 ]	0.0705	+3.69%	0.1716	+4.51%	0.2746	+4.12%

TABLE II : Chesapeake Bay under autoregressive rollout with

T\in\{6,16,30\}

. Imp. is relative to STGNN (Backbone).

Panel of Figure 5 — Fig. 3 : Comparison under autoregressive rollout. Across both directed graphs and unstructured meshes, PRSOBG mitigates boundary-induced instability and achieves consistently lower errors than a vanilla STGNN.

Method	Train (rel.)	Inference (rel.)	Memory (rel.)
STGNN	1.00 $\times$	1.00 $\times$	1.00 $\times$
Ghost-Only	1.01 $\times$	1.01 $\times$	1.01 $\times$
Ours-Explicit	1.04 $\times$	1.03 $\times$	1.02 $\times$
Ours-Convexified	1.08 $\times$	1.08 $\times$	1.05 $\times$

TABLE III : Computational cost relative to the STGNN.

Method	Ghost RMSE	Pearson Corr.
VN	0.1018	0.206
Ghost	0.0705	0.961

TABLE IV : Controlled ghost-proxy validation under known upstream forcing (

T{=}6

Method	RMSE	Imp.	IW	Imp.
STGNN	0.3499	–	0.4423	–
Ghost-Only	0.2777	20.63%	0.2422	45.24%
Ours-Conv.	0.2311	33.95%	0.1770	59.98%

TABLE V : Extreme/high-flow evaluation on River System under autoregressive rollout (

T{=}6

Change	$\lambda_{\rm reg}$	$\Delta t$	$\hat{g}$	RMSE / $\%\Delta$
Default	1.0	1.0	1.0	0.2311 / –
$\lambda_{\rm reg}$ +20%	1.2	1.0	1.0	0.2324 / +0.56%
$\lambda_{\rm reg}$ -20%	0.8	1.0	1.0	0.2322 / +0.47%
$\Delta t$ +20%	1.0	1.2	1.0	0.2339 / +1.21%
$\Delta t$ -20%	1.0	0.8	1.0	0.2309 / -0.09%
$\hat{g}$ +20%	1.0	1.0	1.2	0.2364 / +2.29%

TABLE VI : Sensitivity of the Convexified Refiner.

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COGENT: Continuous Graph Emulators with Neural Ordinary Differential Equations for Long-Term Physical Forecasting

Zesheng Liu, Maryam Rahnemoonfar

Lehigh University · Bethlehem, Pennsylvania, USA

Long-Horizon Forecasting Neural Ordinary Differential Equations Global Climate Models Emulators Time Discretization Ordinary Differential Equations+2

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In this work, we present COGENT, a continuous graph emulator with Neural Ordinary Differential Equations for long-term physical forecasting on irregular geospatial meshes. COGENT encodes a finite history of system states and associated forcing fields and external forcings with a graph-based history encoder, producing node-wise context vectors that capture both local spatial interactions and temporal evolution. These context vectors initialize and condition a latent Neural Ordinary Differential Equation whose dynamics are driven by interpolated future forcings and explicit relative rollout time. By modeling the forecast trajectory as a continuous latent dynamical system, COGENT can generate predictions at arbitrary future times rather than being restricted to a fixed temporal discretization. A residual decoder maps the resulting latent trajectories back to future physical states, enabling direct multi-step forecasting without repeatedly feeding predicted states back into the model. This formulation combines graph-based spatial representation, history-conditioned latent dynamics, and continuous-time rollout in a unified framework for mesh-based physical simulation emulation. In order to stabilize training with long-horizon supervision, we also propose effective rollout-horizon sampling and a progressive rollout-horizon scheduling strategy. We evaluate COGENT on transient ice-sheet simulations generated by the Ice-sheet and Sea-level System Model, demonstrating improved long-range stability over autoregressive graph baselines. These results suggest that continuous graph Neural ODEs provide a promising methodology for scalable physical forecasting on irregular geospatial meshes, particularly in applications that require stable long-horizon predictions and the ability to query system states at arbitrary times.

Abstract

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