No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty

Technical Summary: No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty

Problem Statement
Physics-informed learning, particularly Physics-Informed Neural Networks (PINNs), is increasingly applied to inverse problems governed by partial differential equations (PDEs). While these methods can improve reconstruction quality in familiar regimes, their reliability in ill-posed settings remains difficult to certify. A learned reconstruction may appear visually plausible or achieve low training loss while remaining unstable under data perturbations, model mismatches, or out-of-distribution shifts. Current approaches often lack computable reliability information, making it impossible to distinguish between a genuine improvement and a "hallucinated" or unstable solution. The central challenge is to develop a framework that allows learned solvers to improve upon traditional baselines without silently degrading reliability.

Methodology
The paper proposes a no-harm certification-and-selection framework that acts as a decision layer rather than a new reconstruction architecture. The core principle is that a learned inverse reconstruction should only replace a baseline reconstruction if its residual-calibrated uncertainty radius is demonstrably no worse than that of the baseline.

The methodology proceeds through the following steps:

1. Residual Definition: For a candidate reconstruction $(u, q)$, the framework computes four distinct residual components:

   • Data Residual ($r_{data}$): The mismatch between the model output and noisy observations.
   • Physics Residual ($r_{pde}$): The violation of the governing PDE.
   • Boundary/Initial-Condition Residual ($r_{bc}$): The mismatch with boundary or initial constraints.
   • Optimization Residual ($r_{opt}$): A measure of stationarity (e.g., gradient norm) to account for approximate minimizers and unfinished training.

2. Conditional Stability and Error Bounds: The framework relies on a conditional stability estimate (Assumption 1), which posits that for an admissible set of solutions, the reconstruction error is bounded by a power of the sum of residuals and noise. Under this assumption, the paper derives an a posteriori error bound (Theorem 1). This bound defines a residual-calibrated uncertainty radius ($R$) that serves as a deterministic upper bound on the reconstruction error.

3. Stochastic and Optimization Extensions:

   • For physics residuals estimated via random collocation points, the paper derives high-probability upper bounds (Theorem 3) using Hoeffding's inequality, ensuring the certificate remains valid even when residuals are sampled.
   • An optimization residual is explicitly included to penalize solutions where the optimizer has not reached a meaningful stationary point, preventing the acceptance of "unfinished" training results.

4. The No-Harm Selection Rule: Given a baseline reconstruction with radius $R_{base}$ and a learned reconstruction with radius $R_{learn}$, the learned solution is accepted only if:
   $$R_{learn} \leq R_{base} + \epsilon_{safe}$$
   where $\epsilon_{safe}$ is a prescribed tolerance. If this condition fails, the framework defaults to the baseline. This ensures that the selected output is always controlled by the baseline's reliability certificate.

Key Contributions
The paper makes seven primary contributions:

1. Residual Certificate Definition: A unified certificate combining data, PDE, boundary/initial-condition, and optimization residuals.
2. A Posteriori Error Bound: A proof that, under conditional stability, these residuals yield a computable upper bound on the reconstruction error.
3. Uncertainty Radius: The introduction of a residual-calibrated uncertainty radius that can be reported alongside learned reconstructions.
4. No-Harm Selection Rule: A deterministic rule that accepts learned reconstructions only when they improve upon or match a baseline's certified reliability.
5. Stochastic Certification: High-probability bounds for physics residuals estimated from independent random collocation points.
6. Optimization Residual Integration: A mechanism to account for approximate minimizers and incomplete training via stationarity diagnostics.
7. Numerical Validation: Comprehensive testing across diverse inverse problems (Poisson source recovery, inverse heat reconstruction, limited-angle tomography, and elliptic coefficient identification) to validate the framework's behavior under noise, model mismatch, and ill-posedness.

Results
Numerical experiments validate the framework across five distinct scenarios:

• Poisson Source Recovery: The selector successfully accepted "good" learned candidates with smaller certified radii while rejecting "shifted" and "unfinished" candidates that had larger radii, despite some having plausible visual appearances.
• Inverse Heat Reconstruction: In strongly ill-posed regimes (where the forward map is highly smoothing), the framework became conservative, rejecting learned candidates even when their true error was moderate, because their certified radii exceeded the baseline due to instability.
• Limited-Angle Tomography: A critical stress test showed that a learned candidate with a lower true error (hindsight) was rejected because its certified radius was larger than the baseline. This demonstrates the rule's reliance on available evidence rather than ground truth.
• Geophysical Coefficient Identification: The framework accepted learned improvements in coefficient identification while rejecting shifted candidates, confirming its ability to detect structural inconsistencies.
• Stochastic Residual Validation: Tests with independent collocation points confirmed that the high-probability bounds are conservative and tighten as the number of validation points increases, achieving 100% coverage in the tested regimes.

A large-scale sufficiency sweep (21,000 trials) revealed that the no-harm rule is highly conservative: it rejected over 54% of learned candidates that were actually more accurate than the baseline (false rejections) to ensure that no unsafe candidates (less than 1.2% of selections) were accepted. This confirms the framework prioritizes reliability over maximizing potential accuracy when ground truth is unknown.

Significance and Claims
The paper claims that its primary contribution is not a new neural architecture or a specific solver, but a reusable decision rule for physics-informed inverse learning. The framework shifts the operational paradigm from "reconstruct and hope" to "reconstruct, certify, and select."

The authors emphasize that the method is architecture-independent, capable of wrapping PINNs, neural operators, unrolled solvers, or classical optimizers. The significance lies in providing a mechanism to prevent uncertified learned outputs from silently replacing safer, traditional baselines. By tying acceptance to a residual-calibrated certificate derived from conditional stability, the framework ensures that any improvement in reconstruction is supported by data consistency, physics compliance, and optimization convergence, rather than visual plausibility or training loss alone. The paper concludes that in the absence of ground truth, the no-harm principle provides a necessary safeguard for deploying physics-informed learning in critical inverse problems.