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Physics-Refined Spatiotemporal Forecasting on Open-Boundary Hydrologic Graphs

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

arXiv:2610.01765v1cs.LG

TL;DR

Open-boundary forecasts lack external forcing, and boundary inconsistencies can compound into downstream errors during autoregressive rollout,,,. PRSOBG learns ghost-node proxies and applies local and global physics refiners; across two hydrologic systems, it outperforms 9 baselines with a 38.3% average RMSE reduction over the STGNN backbone across reported horizons.

  • Problem

    Open-boundary forecasting lacks observed external forcing, and small boundary inconsistencies can compound into substantial downstream errors during autoregressive prediction,,,.

    Source: §I · p. 1

  • Method

    PRSOBG learns ghost-node proxies from observable boundary–interior dynamics and refines their propagation with local boundary and global physics refiners.

    Source: §I · pp. 1–2

  • Results

    PRSOBG outperforms 9 baselines, with an average RMSE reduction of 38.3% over the STGNN backbone across reported horizons on two real hydrologic systems.

    Source: §I · p. 2

  • Takeaways & Limitations

    The framework combines learned boundary-forcing proxies with physics-guided corrections to address long-horizon autoregressive instability in open-boundary hydrologic graphs.

    Source: §VI · p. 10

Abstract

from arXiv · show

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.

I. INTRODUCTION

Open-boundary forecasting lacks external forcing that enters through boundary nodes, and small boundary inconsistencies can compound into downstream errors during autoregressive prediction,,,. The paper frames this as a coupled question of what forcing to represent and how to propagate it, then introduces PRSOBG and evaluates it on two hydrologic systems.

I. INTRODUCTION

  • Boundary nodes regulate unobserved external forcing, and autoregressive forecasts can compound small boundary inconsistencies into substantial downstream errors.In the Danube example, boundary nodes constitute 58.4% of the network, and errors increase from upstream to downstream.Source: §I · p. 1, Figure 1 · p. 1
  • The paper asks what unobserved boundary forcing enters the forecast domain and how it should propagate without amplifying errors during autoregressive rollout.These questions organize the paper's analysis of open-boundary forecast instability.Source: §I · p. 1
  • PRSOBG learns ghost-node proxies from boundary and adjacent interior dynamics, then uses local and global physics refiners to stabilize their propagation.The local refiner addresses proxy–boundary error feedback, while the global refiner applies a physics-guided graph correction.Source: §I · pp. 1–2
  • PRSOBG outperforms 9 baselines, with an average RMSE reduction of 38.3% over the STGNN backbone across reported horizons.The baselines comprise 6 spatiotemporal graph-learning models and 3 physics-informed competitors; evaluation uses the Danube river system and Chesapeake Bay.Source: §I · p. 2

II. PRELIMINARIES

  • The setup distinguishes directed river graphs from unstructured coastal meshes and forecasts discharge or Total Water Level, respectively, through autoregressive rollout.The graph formulation uses node features, adjacency, and edge attributes as inputs.Source: §II · p. 2

III. PROPOSED FRAMEWORK

PRSOBG addresses open-boundary instability by learning proxies for missing forcing from in-domain dynamics and refining their propagation. Its proxy is motivated by a discrete boundary relation using the boundary state and its difference from an immediate interior neighbor.

III. PROPOSED FRAMEWORK

  • PRSOBG learns ghost-node proxies from boundary and adjacent interior dynamics, then applies local boundary and global physics refiners during rollout.The global refiner may be explicit or convexified semi-implicit; the local refiner stabilizes proxy–boundary consistency before global correction.Source: §III · p. 3, Figure 2 · p. 3

A. Learn Boundary Forcing via Proxies

  • Because external signals are unavailable at forecast time, the proxy estimates missing boundary forcing from observable boundary and neighboring interior states.For directed rivers, the adjacent in-domain neighbors are immediate downstream nodes; mesh neighbors are one-hop interior nodes.Source: A. Learn Boundary Forcing via Proxies · p. 3
  • The discrete Robin relation motivates a ghost-forcing surrogate parameterized by the boundary state and its difference from an immediate interior neighbor.The one-sided approximation uses the boundary-to-interior state difference to represent the outward normal derivative.Source: A. Learn Boundary Forcing via Proxies · p. 3

B. Ghost Injection under Different Graph Topologies

PRSOBG injects learned ghost proxies according to graph topology: directed graphs use upstream ghost nodes, while unstructured meshes use mesh-specific variants. The mesh variants differ in whether they update boundary states, add mirrored nodes, or learn boundary–ghost coupling.

1) Upstream explicit ghost nodes for directed graphs:

  • Directed graphs receive one explicit upstream ghost node per open-boundary node, with a directed edge that injects forcing along the physical flow direction.Ghost nodes carry boundary forcing through message passing but are excluded from prediction and supervision on the original graph nodes.Source: 1) Upstream explicit ghost nodes for… · p. 4

2) Mesh-specific ghost variants for unstructured meshes:

  • Unstructured meshes use three injection variants: a boundary residual, an explicit mirrored ghost layer, or mirrored ghosts with learnable boundary–ghost coupling.The variants accommodate mesh boundaries that may connect to multiple interior neighbors without a prescribed flow direction.Source: 2) Mesh-specific ghost variants for… · p. 4
  • Learnable boundary–ghost coupling retains a mirrored geometric scaffold while allowing more flexible edge connections.The variant selects candidate ghost nodes by spatial kNN and learns bidirectional gates for boundary–ghost edges.Source: 2) Mesh-specific ghost variants for… · p. 4

C. Boundary Refiner for Proxy Stabilization

The boundary refiner counters proxy errors that could contaminate later proxy construction by enforcing local ghost–boundary–interior consistency during autoregressive rollout. It balances that constraint against the STGNN estimates and applies a closed-form correction to the ghost proxy.

C. Boundary Refiner for Proxy Stabilization

  • Unrefined proxy errors may contaminate subsequent proxy construction and amplify boundary-forcing errors during autoregressive rollout.The local boundary refiner is inserted before the next rollout step.Source: C. Boundary Refiner for Proxy… · p. 4
  • The three-point residual measures the ghost state against a weighted combination of boundary and immediate-interior states.The relation is expressed as r(hg, hb, hn) = hg − αhb − βhn.Source: C. Boundary Refiner for Proxy… · p. 4
  • The refiner balances a three-point consistency constraint against closeness to the backbone’s ghost, boundary, and interior estimates.Its objective is strictly convex and has a unique minimizer.Source: C. Boundary Refiner for Proxy… · pp. 4–5
  • A closed-form update moves the ghost proxy toward local boundary consistency without overwriting the backbone estimate.The correction can be inserted as a single layer at each rollout step.Source: C. Boundary Refiner for Proxy… · p. 5

D. Global Physics Refiner for Boundary-Driven Propagation

A global physics refiner regularizes boundary-driven propagation by applying a structured post-update at each autoregressive rollout step. The corrected representation is used before readout and feedback to reduce drift accumulation without solving the underlying PDE to convergence.

D. Global Physics Refiner for Boundary-Driven Propagation

1) Explicit Refiner:

The explicit refiner uses topology-specific physics priors to shape propagation: an advective prior for directed graphs and a damped-wave prior for unstructured meshes. Its corrected representation feeds prediction readout and autoregressive feedback.

1) Explicit Refiner:

  • For directed graphs, the refiner uses an advective prior with upwind graph derivatives; D1 encodes along-channel spacing and D2 adds elevation-related slope information.The prior is ∂tu + u∂xu = −g0∂xz.Source: 1) Explicit Refiner · p. 5
  • For unstructured meshes, the refiner uses the homogeneous damped-wave equation as a stabilizing propagation prior after treating forcing as absorbed by the data-driven backbone.The prior applies for slowly varying or locally constant depth.Source: 1) Explicit Refiner · p. 5
  • The explicit refiner’s post-update uses current and previous representations alongside adjacency and hidden-space graph-Laplacian terms.The corrected state is used for readout and autoregressive feedback.Source: 1) Explicit Refiner · p. 5

2) Convexified Semi-Implicit Refiner:

The convexified semi-implicit refiner addresses possible mismatch between simplified dynamics and complex open-boundary systems by combining a physics prior with an anchor to the boundary-refined prediction. Its formulations yield strictly positive-definite systems that can be solved stably with PCG.

2) Convexified Semi-Implicit Refiner:

  • The simplified dynamics prior may mismatch complex open-boundary systems, especially on high-resolution graphs or meshes where local errors can amplify into long-horizon drift.The semi-implicit update is convexified into a strictly convex quadratic form, with the boundary-refined prediction as an anchor and the dynamics prior as a structural regularizer.Source: 2) Convexified Semi-Implicit Refiner · p. 5
  • For directed graphs, the semi-implicit update fixes the multiplicative state and evaluates the directed derivative at the updated state.Because the upwind operator is generally non-symmetric, the method solves a convexified objective with graph regularization and damping anchored to the boundary-refined prediction; the resulting system is strictly positive definite and solvable by PCG.Source: 2) Convexified Semi-Implicit Refiner · p. 5
  • For unstructured meshes, the implicit spatial update yields Kb h⋆ = yb, with Kb = I + βL; this system is stable and efficiently solvable when β ≥ 0.The spatial propagation term uses the updated state.Source: 2) Convexified Semi-Implicit Refiner · pp. 5–6

3) Differentiable PCG Layer:

The differentiable PCG layer approximates a unified SPD system for the convexified semi-implicit refiners. It initializes from the boundary-refined representation and uses the approximate solution as the globally refined hidden state for readout and the next autoregressive step.

3) Differentiable PCG Layer:

  • Both convexified semi-implicit refiners are expressed as the unified SPD system K˜h⋆= y, with graph- or mesh-specific K and y.For directed graphs, (K, y) = (Kr, yr); for unstructured meshes, (K, y) = (Kb, yb).Source: 3) Differentiable PCG Layer · p. 6
  • The system solution is approximated with a fixed number of differentiable PCG iterations.Source: 3) Differentiable PCG Layer · p. 6
  • The PCG warm start is the boundary-refined representation, with initial residual r(0) = y − K˜h(0).Source: 3) Differentiable PCG Layer · p. 6
  • A diagonal preconditioner M = diag(m) + εI uses nonnegative parameterized m, and the approximate solution becomes the globally refined hidden state for readout and the next rollout step.Source: 3) Differentiable PCG Layer · p. 6

IV. EXPERIMENTS

Experiments evaluate PRSOBG on a directed river graph and an unstructured coastal mesh under autoregressive rollout. The results examine boundary-error accumulation, ghost proxies, physics refiners, controlled proxy validation, high-flow performance, sensitivity, and computational cost.

Datasets and metrics

  • The benchmarks are Danube discharge forecasting and Chesapeake Bay total-water-level forecasting, evaluated at autoregressive horizons T=6, 16, and 30.Danube has 358 nodes; the Chesapeake mesh has 51,215 nodes and 191 open-boundary nodes.Source: §IV · p. 6

RQ1. Boundary information deficit and error accumulation

  • At T=6, River System STGNN RMSE is 0.3499 versus 0.2623 for GRU, and the backbone error increases to 0.5924 at T=30.The same pattern holds on Chesapeake Bay, where GRU has lower RMSE than the backbone at all reported horizons; Fig. 3 shows PRSOBG has lower errors than vanilla STGNN across both graph types.Source: §IV · pp. 6–7, Figure 3 · p. 6, Table I · p. 7

RQ2. Ghost proxies and boundary refiner

  • On River System, Ghost with Boundary Refiner records RMSE 0.2548, 0.3565, and 0.4679 at T=6, 16, and 30, improving over Ghost-Only at each horizon.On Chesapeake Bay, adding the boundary refiner brings RMSE to 0.0502, 0.1176, and 0.1885; ghost variants also improve over the STGNN backbone.Source: §IV · pp. 7–8, Table I · p. 7, Table II · p. 7

RQ3. Explicit and convexified physics refiners

  • Across both datasets, Ours-Explicit improves over Ghost-Only and Ghost with Boundary Refiner, while Ours-Convexified further improves the reported results.On Chesapeake Bay, Ours-Convexified RMSE is 0.0432, 0.0897, and 0.1398 at T=6, 16, and 30; its improvements over the STGNN backbone are 40.98%, 50.08%, and 51.19%.Source: Table I · p. 7, Table II · p. 7, §IV · p. 8
  • On River System, Ours-Convexified beats PhysicsCorrect at T=6/16/30 (0.2311 vs. 0.2341; 0.3079 vs. 0.3110; 0.4268 vs. 0.4462).On Chesapeake Bay, Ours-Convexified also reduces RMSE versus PhysicsCorrect from 0.0468 to 0.0432, 0.1045 to 0.0897, and 0.1669 to 0.1398 across the same horizons.Source: Table I · p. 7, Table II · p. 7, §IV · p. 8

Computational cost

  • The explicit refiner is more efficient, while the convexified variant is more robust for long rollouts and complex meshes.Ghost fusion operates around open-boundary nodes, the local refiner uses a closed-form update, and the convexified variant uses two fixed PCG steps.Source: §IV · p. 8, Table III · p. 8

RQ4. Controlled validation of ghost proxy

  • With known synthetic upstream forcing, Ghost reduces RMSE from 0.1018 to 0.0705 and raises Pearson Corr. from 0.206 to 0.961 versus Virtual Node.The real-system ghost proxy is a surrogate for missing boundary influence, not an exact recovery of the external process.Source: §IV · p. 8, Table IV · p. 8

RQ5–RQ6. Extreme/high-flow regimes and sensitivity analysis

  • At T=6, Ours-Convexified improves River System RMSE and IW-RMSE over the STGNN backbone by 33.95% and 59.98%, respectively.The sensitivity analysis reports at most 2.29% RMSE variation under perturbed physical initialization and within 1.43% under proxy MLP width and depth changes.Source: §IV · p. 9, Table V · p. 9, Table VI · p. 9

V. RELATED WORK

Prior work spans graph and mesh forecasting, open-boundary treatments, and correction methods for stable rollouts. PRSOBG adapts physics-refinement ideas specifically to unresolved open-boundary forcing.

Hydrologic and flood forecasting on meshes/graphs

  • Graph and mesh forecasting uses architectures such as MeshGraphNets, DCRNN, and Graph WaveNet, alongside models targeting flood dynamics and long rollouts.Examples include FloodGNN-GRU, mSWE-GNN, and HydroGAT.Source: §V · p. 9

Open boundaries in numerical models and learned surrogates

  • Classical open-boundary models use radiation or characteristic conditions, relaxation or nudging, and sponge layers, while many learned surrogates assume boundary signals are available at rollout.When forcing or observations are available, related approaches inject them through data fusion or condition on boundary or discretization information.Source: §V · p. 9
  • Unlike the global virtual node in, PRSOBG uses boundary-specific ghost proxies inferred from local boundary–interior dynamics to approximate unresolved external forcing.Other learned surrogates often assume external boundary signals are available at inference time.Source: §V · p. 9

Physics refiners for stable long rollouts

  • PRSOBG builds on correction methods including PDE-Refiner and PhysicsCorrect, but specifically regularizes propagation under unresolved open-boundary forcing.Related work also includes future-aware objectives, data assimilation methods, and global/local physical-consistency constraints in DUALFloodGNN.Source: §V · pp. 9–10

VI. CONCLUSION

PRSOBG combines ghost-node proxies and physics refiners to address missing boundary forcing and error amplification in long autoregressive rollouts. Experiments on two real hydrologic systems show consistent improvements in rollout accuracy and stability.

VI. CONCLUSION

  • Experiments on two real open-boundary hydrologic systems show consistent improvements in rollout accuracy and stability.The conclusion identifies both systems as real-world hydrologic systems.Source: §VI · p. 10
  • PRSOBG learns ghost-node proxies for unobserved boundary forcing and uses boundary and global physics refiners to regularize local consistency and boundary-driven propagation.The boundary refiner stabilizes proxy–boundary consistency; the global refiner regularizes propagation.Source: §VI · p. 10
  • Because simplified PDE priors may not capture real-world hydrologic complexity, PRSOBG adds a convexified semi-implicit refiner for stable corrective updates.It updates the boundary-refined prediction rather than solving the full PDE to convergence.Source: §VI · p. 10
  • Future work will investigate abrupt external forcing and uncertainty-aware forecasting under such events.Source: §VI · p. 10

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