PDE foundation models enable data-efficient adaptation but are deployed autoregressively, so short-horizon accuracy need not imply reliable simulation. We ask whether the short-horizon advantage of a foundation-model recipe persists when its predictions are repeatedly fed back as inputs for a long horizon. We define error amplification as G=E(H_f)/E(H_s), which measures full-horizon error relative to short-horizon error. We evaluate seven models on seven two-dimensional time-dependent PDE datasets: scratch-trained FNO and DeepONet; fine-tuned DPOT, GPhyT, Poseidon, and Walrus; and frozen in-context VICON. We report VRMSE, relative L_2, and relative MAE. Six datasets provide paired short- and full-horizon scores, while CE-RM contributes short-rollout sweeps only. At least one fine-tuned foundation model outperforms both scratch baselines in 26 of 30 short-rollout dataset--budget cells under every metric. Nevertheless, full-horizon VRMSE exceeds short-horizon VRMSE in 140 of 144 paired trained configurations, with a median error amplification of $2.04$. The count below VRMSE one falls from 104 to 53, and 12 full rollouts exceed ten although none does at the short horizon. Walrus achieves the lowest full-horizon VRMSE in 16 of 24 cells, yet its median error amplification is $2.70$. Relative L_2 and relative MAE show the same qualitative pattern. These results show that favorable short-horizon rankings can coexist with substantial error amplification. PDE foundation-model benchmarks should therefore report horizon-resolved absolute errors, amplification statistics, and failure incidence rather than a single local forecast score.

Autoregressive PDE foundation models enable fast full-field forecasting but accumulate errors as predicted states are recursively reused. We introduce Event-Triggered Context Healing (ETCH), which compares forecasts with measurements from 12 cylinder-surface pressure taps. When their discrepancy exceeds a threshold, pressure-conditioned hourglass diffusion transformers reconstruct the velocity--pressure context for subsequent predictions. On 43 RealPDEBench Cylinder trajectories, ETCH reduces velocity and pressure errors from 48.05 and 63.71% to 6.29 and 7.25% for DPOT, and from 74.97 and 76.91% to 5.16 and 6.84% for Poseidon-T, using the same reconstructors and threshold. For DPOT, correcting 14.56% of windows maintains errors close to reconstructing every window and supports average real-time forecasting and correction on an A100 GPU. These results show that sparse physical feedback stabilizes long autoregressive rollouts across PDE foundation models while reducing reconstruction cost.

Real-time monitoring of safety-critical interior states remains an open problem across energy, environmental, and industrial systems where direct instrumentation is infeasible. Existing approaches based on governing equations, discrete state vectors, or fixed sensor locations fail to provide mesh-independent, field-level reconstructions at arbitrary interior coordinates under real-time constraints. Here we introduce neural operator-based virtual sensing as a general framework for recovering inaccessible interior fields from sparse, heterogeneous measurements and operating conditions, instantiated with MIMONet, a multi-input, multi-output neural operator that fuses heterogeneous inputs and decodes coupled physical fields through a shared latent representation. Across three engineering-grade evaluations of escalating complexity, from confined recirculating flows to pressurized water reactor subchannels and compact power-system heat exchangers, MIMONet achieves ≤5% relative reconstruction error, while generating full-field predictions with sub-millisecond inference on an NVIDIA H200, at least five orders of magnitude faster than traditional high-fidelity CFD simulations, with calibrated uncertainty and noise resilience demonstrated on the heat-exchanger case. We validate the framework on three independent real-world datasets spanning electrochemical energy, atmospheric science, and physical oceanography: current-density mapping in a hydrogen fuel cell, wind-speed profiling on instrumented meteorological towers, and North Atlantic ocean-state from a global ocean reanalysis. MIMONet improves over classical virtual-sensing baselines by 72–84%: 84% on fuel-cell internal-field recovery from 10% sensor coverage and 83% on hub-height wind-speed recovery from lower-level tower sensors. Most notably, it recovers sea-surface height, a field with zero direct sensors, from its learned coupling to observed temperature and salinity at 4.1 × lower error than the best classical predictor, a cross-modal field recovery beyond the reach of classical interpolation. These results establish operator-based virtual sensing as a practical route to real-time field observability in systems where the states that matter most cannot be directly measured. The authors introduce an operator-learning framework for real-time virtual sensing that reconstructs inaccessible interior physical fields from sparse, heterogeneous measurements and operating conditions.

Deep neural operators learn mappings between input functions and complete PDE solution fields, enabling forward evaluations of new problem instances orders of magnitude faster than conventional numerical solvers. Attention mechanisms have recently been introduced into neural operators, but most studies change several architectural components at once, making it difficult to identify what actually improves accuracy. This work presents a controlled and systematic study of five deep operator network (DeepONet) variants with distinct attention mechanisms, trained under both data-driven and physics-informed regimes, to isolate the effects of cross-attention, self-attention, tokenization, and attention depth. We evaluate them on a source-driven transient one-dimensional nonlinear diffusion-reaction equation, a transient one-dimensional viscous Burgers equation with variable initial conditions, and a two-dimensional Poisson heat-conduction problem with heterogeneous source fields. Per-sensor tokenization with cross-attention reduces the mean relative L_2 error of the classical DeepONet in all benchmark-training combinations by factors of 2.4-28.0, while the best attention configurations reach 3.5-32.3. Branch self-attention paired only with dot-product fusion is inconsistent, degrading the one-dimensional problems while helping the more complex two-dimensional source field; added on top of cross-attention it improves all six cases, though by less than cross-attention fusion alone. Global pre-mixing provides no consistent benefit. Increasing cross-attention depth further improves accuracy, but with diminishing returns and a substantially higher cost under physics-informed training. Overall, query-dependent cross-attention is the most reliable mechanism, whereas branch self-attention is most useful for large, spatially complex functional inputs.

Real-time prediction of complex physical systems requires surrogate models that learn from data while representing strong multiphysics coupling. Deep Operator Networks have shown success in single-physics problems, yet their effectiveness in capturing nonlinear interactions in coupled systems (such as thermo-mechanical or electro-thermal coupling) remains underexplored. Here we pose a practical question: should the architecture of a neural operator reflect the strength of physical coupling it aims to model? We compare single-branch and multi-branch designs, in both feedforward and sequential recurrent forms, across three representative systems: a reaction-diffusion problem with heterogeneous sources, a nonlinear thermo-electrical problem with temperature-dependent conductivity and Joule heating, and a viscoplastic thermo-mechanical model of steel solidification. Single-branch networks consistently outperform multi-branch variants in tightly coupled regimes by encouraging shared latent representations, whereas multi-branch designs remain favorable for decoupled or single-physics tasks. Once trained, these surrogates deliver full-field predictions up to 1.8 × 104 times faster than physics-based solvers. Neural operators offer fast surrogates for costly physics simulations. Jaewan Park, Kazuma Kobayashi and colleagues report that single-branch networks are more accurate for tightly coupled physics, while multi-branch designs suit decoupled problems.