TWIG: A Time-Causal Wavelet Operator for Autoregressive Forecasting on Irregular Graphs

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.

Electricity forecasting often involves spatially related signals observed over regions, substations, and feeders, and Graph Neural Networks (GNNs) provide a natural way to represent these relations. Building a complete GNN forecasting experiment is nonetheless laborious, because graph construction, model selection, training, aggregation, and interpretation sit in incompatible tools. We present GraphToolbox, an open-source Python framework that unifies these stages in one configurationdriven pipeline built on PyTorch Geometric. It offers data-driven graph construction, an adapter that instantiates and trains 51 of the 65 PyTorch Geometric convolutions together with the recurrent cells of PyTorch Geometric Temporal, online expert aggregation, forecasting interpretability, and significance testing on cached forecasts. We evaluate the pipeline in two case studies. On French regional load, the 48 convolutions included in the complete forecasting sweep fall in a band from 1.14% to 1.60% error, online aggregation lowers this to 0.98%, and the graph models improve on classical additive and boosting baselines. On net-load, direct graph models are less accurate than a classical additive model, while forecasting each physical component separately improves them without closing that gap. Both comparisons use the same experimental interface, illustrating the role of GraphToolbox in systematic architectural evaluation.

Fast surrogate modeling for high-dimensional physical dynamics requires more than low short-term error: useful models must roll out efficiently while preserving the statistical structure of long trajectories. Neural operators provide inexpensive autoregressive forecasts but can drift in turbulent regimes, whereas rolling diffusion and latent generative surrogates can represent stochastic transitions at the cost of multi-step denoising, noise-schedule design, or auxiliary compression models. We propose MeanFlow Long-term Invariant Spatiotemporal Consistency Autoregressive Models (MeLISA), a latent-free autoregressive generative surrogate built on pixel-space MeanFlow. MeLISA defines a blockwise stochastic transition kernel that generates each forecast block with a single model evaluation, avoiding latent encoders and iterative diffusion solvers at inference time. To stabilize long-horizon rollouts, MeLISA combines a Window-Consistency MeanFlow objective that learns conditional spatiotemporal generation from partially observed temporal windows with a Time Increment Consistency loss that constrains multi-lag finite increments and targets temporal-correlation structure. We evaluate MeLISA with compact UNet and scalable DiT backbones on two high-resolution benchmarks, extended 2D Kolmogorov flow at $256 \times 256$ and turbulent channel-flow slice at $192 \times 192$. MeLISA outperforms neural-operator baselines on short-term forecasting accuracy and long-horizon statistical metrics, including energy spectra, turbulent kinetic energy, and mixing-rate-related dynamics, while achieving inference speeds comparable to, and in some cases faster than, neural operators. Compact 3.7-5.7M-parameter variants already deliver strong parameter efficiency, and DiT variants provide a scalable path up to 150M parameters. Overall, MeLISA benefits both rollout efficiency and long-horizon statistical accuracy.