Cross-Domain Long-Term Forecasting: Radiation Dose from Sparse Neutron Sensor via Spatio-Temporal Operator Network

TL;DR

This work tackles cross-domain, long-horizon forecasting of unobservable radiation-dose fields from sparse ground-based neutron measurements by introducing STONe, a non-autoregressive neural operator. STONe learns a direct operator $\mathcal{G}$ that maps sparse sensor histories to dense spatiotemporal dose fields via a branch/trunk DeepONet-inspired architecture, enabling 180-day forecasts with millisecond inference. Key findings show that memory-enabled temporal encoders (GRU/Transformer) yield superior long-horizon accuracy and stability, with GRU delivering the best average metrics across the horizon, while maintaining real-time inference suitable for operational use. The approach demonstrates a general cross-domain operator learning paradigm applicable to physics, climate, and energy systems, offering real-time, cross-domain reconstruction from sparse observations and opening avenues for uncertainty quantification and multi-physics extensions.

Abstract

Forecasting unobservable physical quantities from sparse, cross-domain sensor data is a central unsolved problem in scientific machine learning. Existing neural operators and large-scale forecasters rely on dense, co-located input-output fields and short temporal contexts, assumptions that fail in real-world systems where sensing and prediction occur on distinct physical manifolds and over long timescales. We introduce the Spatio-Temporal Operator Network (STONe), a non-autoregressive neural operator that learns a stable functional mapping between heterogeneous domains. By directly inferring high-altitude radiation dose fields from sparse ground-based neutron measurements, STONe demonstrates that operator learning can generalize beyond shared-domain settings. It defines a nonlinear operator between sensor and target manifolds that remains stable over long forecasting horizons without iterative recurrence. This challenges the conventional view that operator learning requires domain alignment or autoregressive propagation. Trained on 23 years of global neutron data, STONe achieves accurate 180-day forecasts with millisecond inference latency. The framework establishes a general principle for cross-domain operator inference, enabling real-time prediction of complex spatiotemporal fields in physics, climate, and energy systems.

Figures (6)

• Figure 1: Conceptual framework and STONe forecasting approach for data-driven radiation dose reconstruction. The figure illustrates the transition from a traditional physics-based forward model to a learned inverse operator enabling real-time global dose forecast. a) Forward problem (physics-based simulation): Unobservable latent drivers, primarily the primary cosmic-ray flux ($\Phi$), interact with Earth’s atmosphere and geomagnetic field. High-fidelity Monte Carlo transport is necessary to model the ensuing particle cascades, resulting in sparse ground-based neutron monitor time series ($\mathbf{Y}$) and high-altitude radiation dose fields ($\mathbf{X}$). Evaluating the latent physical operator $\mathcal{H}_{\text{phys}}$ incurs substantial computational latency. b) Inverse problem (learned spatiotemporal operator): A neural operator $\mathcal{H}$ maps observable sensor time series ($\mathbf{Y}$) and spatial query coordinates ($\mathbf{r}$) directly to reconstructed, high-resolution dose fields $\mathbf{X}$, enabling fast, resolution-agnostic inference. c) Single-step prediction (one-step ahead): Given a sequence of inputs $\mathbf{Y}_{1:T}$ and spatial queries $\mathbf{r}$, a predictor outputs the next dose field $\mathbf{X}_{t+1}$. d) Full-rollout forecasting (this work): A non-autoregressive operator predicts the entire future sequence $\mathbf{X}_{1:K}$ from $(\mathbf{Y}, \mathbf{r})$ in a single pass (solid arrows), supporting stable, long-range, full-field reconstruction.
• Figure 2: STONe Architecture for Spatiotemporal Forecasting. The modular design integrates temporal and spatiotemporal encoders where the branch network captures historical dynamics through sequence processing models, while the trunk network generates coordinate-dependent basis functions. The architecture enables principled forecasting by learning the mapping from sparse sensor histories to dense future field reconstructions. (a) Branch Network Temporal Encoder. The branch network processes sequential sensor input history $\bm{u}_{\text{hist}} \in \mathbb{R}^{K_{\text{hist}} \times N}$ through temporal encoders (GRU, LSTM, or Transformer) to extract dynamical state features and produce temporal coefficient vector $b(\bm{u}_{\text{hist}}) \in \mathbb{R}^q$. (b) Trunk Network Spatiotemporal Decoder. The trunk network maps spatiotemporal coordinates $(\bm{r})$ through fully connected layers to generate basis functions $T(\bm{r}) \in \mathbb{R}^{q \times p \times K_{fut}}$, representing the evaluation of $q$ basis functions across $p$ spacial points and $K_{fut}$ future time points. (c) Sequential Data Processing Pipeline. Historical sensor measurements from $K_{\text{hist}}$ past time steps are fed into the branch network's temporal encoder, which captures complex temporal dependencies and system dynamics to encode the current dynamical state. (d) Future State Reconstruction (Output). The STONe combines temporal coefficients from the branch network with spatiotemporal basis functions from the trunk network to reconstruct the complete future state sequence over $K_{\text{fut}}$ time steps through tensor contraction.
• Figure 3: Qualitative effective dose comparison of multi-day forecasting performance for the four STONe branch architectures against ground truth. Each row corresponds to a different forecast lead time: Day 1 (row 1), Day 60 (row 2), Day 120 (row 3), and Day 180 (row 4). The last column shows the ground truth dose field, followed by predictions from each model architecture.
• Figure 4: Mean Absolute Percentage Error (MAPE) comparison of multi-day forecasting performance for the four STONe branch architectures against ground truth. Each row corresponds to a different forecast lead time: Day 1 (row 1), Day 60 (row 2), Day 120 (row 3), and Day 180 (row 4). The results show the MAPE against the ground truth at each forecasting lead time.
• Figure 5: Case Study: Radiation Dose Forecasting for the North American Region. This figure presents a detailed performance evaluation of the four STONe branch architectures for a 180-day forecast. (a) Qualitative comparison of the predicted effective dose fields from the FCN, LSTM, GRU, and Transformer models against the ground truth (GT) at forecast lead times of 1, 60, 120, and 180 days. All plots in this panel share a color scale. (b-d) Quantitative error analysis showing the Mean Absolute Percentage Error (MAPE) for each architecture at (b) Day 1, (c) Day 60, and (d) Day 180. The color scales in the error plots are independent for each lead time to better visualize the relative performance differences between models.