Debiased neural operators for estimating functionals

Abstract: Neural operators are widely used to approximate solution maps of complex physical systems. In many applications, however, the goal is not to recover the full solution trajectory, but to summarize the solution trajectory via a scalar target quantity (e.g., a functional such as time spent in a target range, time above a threshold, accumulated cost, or total energy). In this paper, we introduce DOPE (debiased neural operator): a semiparametric estimator for such target quantities of solution trajectories obtained from neural operators. DOPE is broadly applicable to settings with both partial and irregular observations and can be combined with arbitrary neural operator architectures. We make three main contributions. (1) We show that, in contrast to DOPE, naive plug-in estimation can suffer from first-order bias. (2) To address this, we derive a novel one-step, Neyman-orthogonal estimator that treats the neural operator as a high-dimensional nuisance mapping between function spaces, and removes the leading bias term. For this, DOPE uses a weighting mechanism that simultaneously accounts for irregular observation designs and for how sensitive the target quantity is to perturbations of the underlying trajectory. (3) To learn the weights, we extend automatic debiased machine learning to operator-valued nuisances via Riesz regression. We demonstrate the benefits of DOPE across various numerical experiments.

• The paper introduces DOPE, a one-step Neyman-orthogonal estimator that removes the first-order plug-in bias in estimating nonlinear scalar functionals of neural operator outputs.
• It leverages the Riesz representer and automatic debiased machine learning to achieve quadratic error scaling and √n-consistency under mild conditions.
• Empirical results show significant RMSE reductions and improved robustness across applications, including pharmacokinetics and PDE simulations.

Debiased Neural Operators for Estimating Nonlinear Functionals

Problem Setting and Bias in Plug-in Estimation

Neural operators are widely adopted for learning solution maps of parametric PDEs, providing mesh-independent approximations that generalize across varying inputs. However, in practical applications—e.g., medicine, climate science, and fluid dynamics—the principal quantity of interest is often not the full solution trajectory but a nonlinear scalar summary (functional) such as time above threshold or total energy.

The prevalent plug-in estimation approach computes the functional on the neural operator’s prediction $\hat{S}(A)$: $g(\hat{S}(A))$. This strategy suffers from systematic first-order plug-in bias, especially prominent for nonlinear functionals. The leading error term propagates linearly through the Fréchet differential: $$\mathbb{E}[g(\hat S(A))] - \mathbb{E}[g(S_0(A))] \approx \mathbb{E}[\mathrm{D}g_{S_0(A)}(\Delta(A))]$$ where $\Delta = \hat S - S_0$. This bias does not vanish unless the neural operator error decays at $o(n^{-1/2})$ rates, which is unrealistic in practice. Thus, na\"ive plug-in estimation of functionals from neural operator output is fundamentally inconsistent for statistical inference.

The DOPE Estimator: Orthogonal Debiasing via Riesz Representers

To address this, the paper introduces DOPE, a one-step Neyman-orthogonal estimator that yields debiased inference for scalar nonlinear functionals of neural operator outputs. The key insight is to treat the neural operator as a high-dimensional nuisance parameter and remove the leading bias term through influence-function-based correction. The correction leverages the Riesz representation of the Fréchet derivative: $$\mathrm{D}g_{u}(h) = \langle w_g(u), h \rangle_\mathcal{U}$$ for $u \in \mathcal{U}$.

DOPE achieves debiasing by estimating a weighting function $\beta_0(A)(x)$ (a Riesz representer under the actual observation design) such that, for any $h$,

$$\mathbb{E}[\beta_0(A)(X) h(X) | A] = \mathrm{D}g_{S_0(A)}(h).$$

This allows reconstructing the bias term from observable pointwise residuals, even under irregular, input-dependent measurement protocols.

The resulting estimator is: $g(\hat{S}(A))$0 where $g(\hat{S}(A))$1 denotes partial, possibly irregular, observations per sample.

Theoretical Guarantees

The central theoretical contribution is the establishment of second-order bias properties. Under standard regularity and cross-fitting, the bias of DOPE is

$g(\hat{S}(A))$2

with no first-order term. This is in stark contrast to the plug-in estimator, which exhibits first-order bias linear in the neural operator error.

As a result, $g(\hat{S}(A))$3-consistent inference is possible under $g(\hat{S}(A))$4 rates for the nuisance estimators, far less stringent than classical plug-in requirements. Asymptotic normality and validity of confidence intervals follow under mild conditions.

Learning the Correction: Automatic Debiasing for Operator-valued Nuisances

Practical implementation of DOPE requires estimation of the Riesz weight $g(\hat{S}(A))$5 without closed-form inversion of the possibly input-dependent (and often singular) sampling design. The authors propose a novel extension of automatic debiased machine learning (DML) to operator-valued nuisances, recasting the problem in terms of a population quadratic objective: $g(\hat{S}(A))$6 This variational characterization can be robustly optimized from data, sidestepping unstable density estimation and inversion.

When the Fréchet derivative structure is known (e.g., for linear and polynomial-energy functionals), the Riesz representer can be substituted in a structured parameterization of $g(\hat{S}(A))$7, reducing statistical burden and improving numerical stability.

Empirical Evaluation

The authors conduct extensive experiments on medical pharmacokinetics and PDE benchmarks. Across diverse setups—coverage functionals in pharmacokinetics, spatial thresholds in Darcy flow simulations, and under partial, input-dependent, and irregular observations—DOPE consistently outperforms the plug-in estimator. The RMSE reductions frequently exceed 15–40% relative to the plug-in approach, especially for complex or highly nonlinear functionals.

Further, robustness analyses demonstrate that DOPE’s estimation error scales quadratically, not linearly, with nuisance function error, confirming the theoretical second-order remainder. The prediction-powered inference (PPI) extension of DOPE shows additional sensitivity to fully unlabeled data, yielding greater improvements than the plug-in baseline when leveraging large unlabeled datasets.

Implications and Future Directions

The DOPE framework provides a general recipe for semiparametric inference on nonlinear functionals of high-dimensional models with operator-valued nuisances. Decoupling inference from high-accuracy operator learning is particularly relevant in scientific applications where solution trajectories are not the end product. The proposed automatic DML extension to operators, via primal representations and Riesz regression, opens avenues for efficient debiasing in deeply nested and infinite-dimensional settings.

The approach is broadly architecturally agnostic, integrating seamlessly with any neural operator model, and is inherently able to handle irregular and partial observation patterns. The framework also supports advanced inferential objectives (e.g., PPI) and is compatible with recent developments in doubly robust and orthogonal estimation.

In the context of scientific machine learning, this enables reliable and statistically efficient estimation of summary quantities from learned surrogates, advancing trustworthiness and general applicability. The principled treatment of bias propagation in nonlinear functionals of learned operators is likely to influence future theoretical and practical advances in neural operator theory, operator-valued machine learning, and nonparametric inference.

Conclusion

This work establishes that plug-in estimation for nonlinear functionals of neural operator outputs is generically inconsistent. The proposed DOPE estimator provides a theoretically founded, statistically efficient, and practically stable solution to this challenge, generalizing orthogonal estimation and DML to operator-valued settings. Empirical evidence attests to its effectiveness and robustness. These results provide a foundation for future work on principled inference for complex functionals and introduce scalable automatic debiasing methodologies into operator learning.