Beyond Uniform Sampling: Synergistic Active Learning and Input Denoising for Robust Neural Operators

Operator learning models are rapidly emerging as the predictive core of digital twins for nuclear and energy systems, promising real-time field reconstruction from sparse sensor measurements. Yet their robustness to adversarial perturbations remains uncharacterized, a critical gap for deployment in safety-critical systems. Here we show that neural operators are acutely vulnerable to extremely sparse (fewer than 1% of inputs), physically plausible perturbations that exploit their sensitivity to boundary conditions. Using gradient-free differential evolution across four operator architectures, we demonstrate that minimal modifications trigger catastrophic prediction failures, increasing relative $L_2$ error from $\sim$1.5% (validated accuracy) to 37-63% while remaining completely undetectable by standard validation metrics. Notably, 100% of successful single-point attacks pass z-score anomaly detection. We introduce the effective perturbation dimension $d_{\text{eff}}$, a Jacobian-based diagnostic that, together with sensitivity magnitude, yields a two-factor vulnerability model explaining why architectures with extreme sensitivity concentration (POD-DeepONet, $d_{\text{eff}} \approx 1$) are not necessarily the most exploitable, since low-rank output projections cap maximum error, while moderate concentration with sufficient amplification (S-DeepONet, $d_{\text{eff}} \approx 4$) produces the highest attack success. Gradient-free search outperforms gradient-based alternatives (PGD) on architectures with gradient pathologies, while random perturbations of equal magnitude achieve near-zero success rates, confirming that the discovered vulnerabilities are structural. Our findings expose a previously overlooked attack surface in operator learning models and establish that these models require robustness guarantees beyond standard validation before deployment.

Neural operators are widely used as fast surrogates for numerical PDE solvers, mapping input functions to solution functions. However, their generalizability and robustness are not yet clearly defined in the operator-learning setting, which differs from traditional adversarial robustness definitions. This paper studies the generalizability and robustness of a learned neural operator from a solver-integrated perspective, addressing the challenge that the output of a learned operator and a numerical solver tends to change in tandem under input perturbation. First, we formalize the definition of generalization and robustness through a model-solver error operator, identifying fixed-input model-solver loss as generalization metric, and norm-bounded adversarial attack loss increase and Jacobian-error function norm as robustness metric. Second, we identify the solver-integrated adversarial attack as appropriate for PDE operator learning and show why model-only or fixed-ground-truth attacks can be insufficient when the solver output also changes with the input. Third, we develop solver-integrated adversarial training methods for neural operators. Experiments on representative PDE benchmarks show that this solver-integrated adversarial training clearly improves both generalizability and robustness. Deeper solver integration yields more effective attacks, more informative samples, and more efficient training than less integrated alternatives. These results provide a general framework for robust operator training and automatic sample selection without heavy manual intervention. More broadly, the formulation applies to adversarial regression whenever a ground-truth oracle can evaluate, and ideally differentiate, the true input-output map; PDE operator learning is one such case.

Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics-based acquisition - a novel physics-informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier-Stokes equations. We show that, in our experiments, physics-based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.