Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

Technical Summary: Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

Problem Statement
In computational science and engineering, inverse problems involve inferring model parameters from observations. A critical, often overlooked step is assessing whether a fitted model is adequate for prediction. Standard practice relies on residual-norm diagnostics, such as Morozov's discrepancy principle or fixed-sample $\chi^2$ tests, which check if the magnitude of residuals is consistent with the assumed noise level.

The paper argues that these magnitude-based diagnostics are misleading in smoothing Partial Differential Equation (PDE) inverse problems. In such problems, structured model errors (misspecification) can be attenuated by the forward operator in the observation space. Consequently, the residual norm may remain below practical discrepancy thresholds even while coherent, systematic residual patterns persist. This leads to the acceptance of fitted models that yield biased parameters and materially wrong quantities of interest (QoIs), despite appearing "consistent" with noise.

Methodology
The authors propose a structure-sensitive sequential diagnostic based on e-processes from game-theoretic statistics. The method operates as a post-fit diagnostic on a fixed parameter estimate $\hat{\theta}$.

1. Expert Portfolio: The method employs a "bank" of experts, where each expert tests for a specific spatial residual pattern. These patterns are drawn from a dictionary of smooth, low-frequency basis functions (Fourier modes and polynomials) relevant to the PDE's geometry.
2. E-Process Construction: Each expert acts as a likelihood-ratio bet against the null hypothesis that residuals are pure noise ($N(0, \sigma^2)$). The experts are combined into a universal portfolio (a mixture of likelihood ratios).
3. Sequential Monitoring: As observations arrive, the portfolio's "wealth" (the aggregate likelihood ratio) is updated. The diagnostic rejects the fitted model if the aggregate wealth crosses a prescribed threshold ($1/\alpha$).
4. Anytime-Valid Control: Leveraging the properties of e-processes and Ville's inequality, the method guarantees anytime-valid type-I error control. This means the analyst can inspect the diagnostic repeatedly, stop data collection early, or continue collecting data based on preliminary evidence without inflating the false-positive rate.
5. Attribution: Upon rejection, the distribution of wealth across the expert portfolio attributes the evidence to specific spatial residual patterns, offering interpretable guidance for model correction.

Key Contributions
The paper makes three primary contributions:

1. Formulation: It formulates a portfolio e-process for residual-pattern detection that provides anytime-valid early warnings for a fixed fitted model.
2. Attribution: It demonstrates how portfolio weights provide interpretable attribution to spatial residual patterns, suggesting parsimonious follow-up corrections.
3. Empirical Demonstration: It shows across three distinct PDE inverse problems that standard discrepancy checks can accept misspecified models with small but structured residuals, leading to biased QoIs, whereas the proposed method detects these failures earlier and with structure sensitivity.

Experimental Results
The method was evaluated on three inverse problems:

• 1D Poisson Diffusivity: A controlled synthetic problem where misspecification types (piecewise, Gaussian bump, three-step, linear) were varied.
  • Result: In the "linear" case, the model error was sub-noise ($\|\mu^*\| < \sigma$). Morozov's check and fixed-sample $\chi^2$ tests failed to detect the error (or required the full dataset), while the e-process detected it sequentially using only 9% of the data. Crucially, the accepted model produced QoI errors of up to 15.3% (point prediction) and 7.8% (regional average).
• 2D Stokes Flow (Basal Drag): An ill-posed boundary inference problem where base-level errors are smoothed to the surface.
  • Result: A misspecified model ($K=3$ vs. true $K=4$) produced surface residuals consistent with noise (RMS/$\sigma \approx 1.35$). Morozov accepted the fit, but the e-process detected the structural inadequacy after only 10% of observations. The inferred basal drag was qualitatively wrong, displacing the minimum drag location critical for grounding-line stability.
• Icepack (Glaciological Ice-Stream): A community finite-element model using the Shallow Shelf Approximation (SSA).
  • Result: Similar to Stokes, the e-process detected structural misspecification ($K=3$ vs. $K=4$) while Morozov accepted the fit. The accepted model resulted in a 36% error in the predicted terminus ice flux.
• Model Repair: In the Poisson experiments, the wealth distribution successfully identified the dominant residual modes. Using these top modes to fit a discrepancy correction reduced QoI errors from ~15% to ~3-4%, matching the performance of a blind regression on the full basis.

Significance and Claims
The paper claims that the e-process offers a practical workflow for detecting residual structure during data acquisition, a capability lacking in standard batch or magnitude-based tests. Its significance lies in:

• Early Detection: It identifies misspecification from a fraction of the observations, often before the residual magnitude becomes large enough to trigger standard checks.
• Interpretability: Unlike black-box rejection, the wealth distribution identifies which spatial patterns drive the lack of fit, guiding exploratory model correction.
• Robustness: The method is computationally efficient (post-processing step, $O(K)$ per observation) and requires no modification to the forward solver or retraining.
• Validity: It provides rigorous type-I error control under optional stopping, making it suitable for settings where diagnostics are inspected repeatedly or data collection is adaptive.

The authors note limitations: the method assumes a fixed $\hat{\theta}$ (conditional validity) and known noise level $\sigma$. While robust to overestimation of noise, underestimation breaks the validity guarantee. The framework is designed for post-fit diagnostics on point estimates, not for fully online refitting of parameters, which would require additional safeguards like universal inference.