Operator Accuracy (OA) in PDE Surrogates

• Operator Accuracy (OA) is a metric that measures the relative error between surrogate operators and high-accuracy numerical or analytical references in function spaces.
• It is widely used in neural operator surrogates, physics-informed networks, and operator preconditioning to benchmark and improve solution fidelity.
• OA enables practical diagnostics such as residual monitoring and adaptive error estimation, which are critical for reliable optimization and dynamic simulations.

Operator accuracy (OA) is a quantitative metric that evaluates the fidelity of an approximate solution operator—whether obtained via neural operators, physics-informed networks, or discretized PDE frameworks—against a high-accuracy numerical or analytic reference. OA provides a precise measure of how well an operator surrogate, typically constructed for tasks such as parametric PDE solving, forecasting time-dependent dynamics, or accelerating optimization, approximates the true mapping at the level of function spaces rather than individual scalar evaluation. OA metrics underpin the rigorous benchmarking, reliability assessment, and error diagnosis of operator learning methodologies across computational science and engineering contexts.

1. Mathematical Definition and Context

Let $\mathcal{F}: \mathcal{M} \to V$ be the exact solution operator for a parameter-dependent boundary value problem, mapping a parameter field $m \in \mathcal{M}$ to the corresponding state $u \in V$ (e.g., $V=H^1_0(\Omega)$). For a surrogate operator $\mathcal{F}_\text{approx} : \mathcal{M} \to V$—such as a neural operator, reduced-order model, or corrected surrogate—OA is defined for each parameter $m$ by

$$\mathrm{OA}(m) := \frac{\| \mathcal{F}_\text{approx}(m) - \mathcal{F}(m) \|_V}{\| \mathcal{F}(m) \|_V} .$$

The choice of $V$ and its norm (e.g., energy norm, $L^2$ norm) depends on the problem structure and intended application.

This definition applies equally to problems formulated in the time domain for evolution equations. For data-driven time-derivative operator learning, OA may measure relative error in operator output (e.g., predicted time derivatives), adopting the form

$$\mathrm{OA}(n) := \frac{ \| \widetilde{\mathbf u}_t^n - \mathbf u_t^n \|_2 }{ \| \mathbf u_t^n \|_2 }$$

where $m \in \mathcal{M}$0 is the learned derivative and $m \in \mathcal{M}$1 the high-resolution reference at time step $m \in \mathcal{M}$2 (Mandl et al., 7 Aug 2025).

2. Operator Accuracy in Neural Operator Surrogates

Neural operator surrogates for parametric PDEs frequently require robust accuracy metrics for error quantification and to guarantee reliability in downstream inference, optimization, and control. The framework in (Jha, 2023) considers OA in the finite element energy norm: $m \in \mathcal{M}$3 where $m \in \mathcal{M}$4 is the neural operator prediction, $m \in \mathcal{M}$5 is the finite element reference, and $m \in \mathcal{M}$6 is a corrected prediction via the Residual-Based Error Corrector Operator.

OA permits evaluation over distributions of problems (e.g., $m \in \mathcal{M}$7), with statistics such as

$m \in \mathcal{M}$8

reported to capture both average-case and worst-case surrogate fidelity.

Numerical results attest to the sensitivity of OA to surrogate correction: mean uncorrected OA may range from approximately $m \in \mathcal{M}$9–$u \in V$0, whereas residual-based correction yields $u \in V$1–$u \in V$2—a two order of magnitude improvement (Jha, 2023).

3. Operator Accuracy in Physics-Informed Operator Learning

OA is central to benchmarking in physics-informed operator learning, especially for time-dependent PDEs. In the PITI-DeepONet formulation (Mandl et al., 7 Aug 2025), OA quantifies the difference between the predicted and reference time-derivative operators: $u \in V$3 Benchmarking across classical evolution equations (1D heat, 1D Burgers, 2D Allen–Cahn), OA as measured by mean relative $u \in V$4 error at final inference times demonstrated substantial improvements: up to $u \in V$5 versus full rollout (FR) and $u \in V$6 versus autoregressive (AR) baselines (see Table 1).

Stability and error accumulation in long rollouts are directly linked to OA. Consistency losses that align operator outputs and auto-differentiation reconstructions further enhance OA and downstream stability, as shown by sustained low error far beyond the training window (Mandl et al., 7 Aug 2025).

PDE/Method	FR	AR	PITI-RK4
Heat 1D (mean OA)	1.5	1.2	0.24
Burgers 1D (mean OA)	0.13	0.94	0.017
Allen–Cahn 2D (mean OA)	0.20	1.1	0.12

4. OA, Preconditioning, and Numerical Linear Algebra

Operator accuracy is not limited to neural or physics-informed operator learning. The context of full operator preconditioning (FOP) (Mohr et al., 2021) reveals that solution OA for discretized operator equations is a function of the conditioning of the discretized operator. FOP preprocesses at the operator level (applying operators $u \in V$7 before discretization) to produce discrete systems with $u \in V$8, attaining $u \in V$9 accuracy, as opposed to standard matrix preconditioning which cannot improve the fundamental $V=H^1_0(\Omega)$0 error bound.

The dependence is quantified precisely: $V=H^1_0(\Omega)$1 where $V=H^1_0(\Omega)$2 is the Galerkin matrix for the bilinear form, $V=H^1_0(\Omega)$3 the boundedness constant, $V=H^1_0(\Omega)$4 the coercivity constant, and $V=H^1_0(\Omega)$5 the Gram matrix condition number (Mohr et al., 2021).

OA as attainable numerical solution fidelity is thus maximized by controlling operator-level conditioning prior to discretization, as evidenced in Chebyshev interpolation and advanced spectral and FEM solvers.

5. Statistics, Monitoring, and Adaptivity via OA

The practical deployment of OA measurements allows systematic surrogate evaluation and online diagnostics. Error statistics over $V=H^1_0(\Omega)$6 support robust uncertainty quantification and estimator calibration (Jha, 2023, Mandl et al., 7 Aug 2025).

In online inference, OA-driven residual monitoring acts as an adaptive quality estimator. The field residual,

$V=H^1_0(\Omega)$7

shows near-perfect Pearson correlation with squared field error ($V=H^1_0(\Omega)$8), enabling detection of out-of-distribution states, dynamic step adjustment, or active-learning triggers (Mandl et al., 7 Aug 2025). This self-diagnostic property extends operator accuracy beyond offline benchmarking to real-time reliability enhancement.

6. OA in Optimization and Cost–Accuracy Trade-Offs

The impact of OA on downstream applications is manifest in optimization workflows involving surrogate models. In constrained topology optimization with neural operator surrogates, uncorrected surrogates yielded optimizer errors up to $V=H^1_0(\Omega)$9, while OA-improved predictions via correction restricted optimizer error below $\mathcal{F}_\text{approx} : \mathcal{M} \to V$0 across all tested networks (Jha, 2023).

Correction frameworks add only minimal computational overhead (a single sparse linear solve per inference), yielding two orders of magnitude gain in OA at a negligible additional cost relative to direct nonlinear PDE solves. This enables the deployment of operator surrogates with confidence in high-stakes inference, design, and control tasks.

7. Summary and Scope

Operator Accuracy (OA) serves as a rigorous, function-space-level metric for surrogate fidelity, underpinning both theoretical analysis and empirical validation across neural operators, physics-informed networks, and classical numerical PDE approximations. Accurate quantification and improvement of OA through correction schemes, residual monitoring, and preconditioning are central to producing reliable, high-precision solutions for parametric and time-dependent PDEs and for enabling robust downstream applications. OA thus occupies a foundational position in modern scientific computing and operator learning frameworks (Jha, 2023, Mandl et al., 7 Aug 2025, Mohr et al., 2021).