Technical Summary: Counterfactual Operator Relevance for PDE Discovery

1. Problem Statement

Data-driven discovery of Partial Differential Equations (PDEs) often relies on sparse regression methods (e.g., SINDy, PDE-FIND) to select active terms from a candidate library. A persistent limitation of these approaches is the conflation of residual fit with functional necessity. A term may receive a non-zero coefficient or reduce the residual simply because it is correlated with other features, the trajectory excites only a narrow subspace, or derivative estimates are noisy. Consequently, residual magnitude alone is an unreliable proxy for the structural importance of an operator.

Furthermore, standard discovery methods often fail to account for:

• Operator Aliasing: When different operator combinations produce identical residuals on a specific dataset (e.g., single-mode trajectories).
• Constraint-Null Non-identifiability: When operators vanish on the admissible constraint class of the data (e.g., divergence-free fields), making their coefficients unidentifiable.
• Lack of Causal Diagnostics: Existing methods do not rigorously distinguish between terms that merely improve a fit and terms that are functionally necessary to reproduce the system's dynamics.

2. Methodology: Counterfactual Operator Interventions

The paper proposes a framework based on counterfactual operator interventions. Instead of relying solely on coefficient magnitude, the method tests whether a candidate term is functionally necessary by comparing the "factual" trajectory (the fitted model) with a "counterfactual" trajectory (the model with the term deleted or perturbed).

Core Definitions

• Intervention: Replacing a candidate operator $T_j$ or its coefficient $\alpha_j$ with a modified version (e.g., deletion).
• Counterfactual Response: The deviation $\delta_j = \|u - u^{cf}_j\|$ between the factual solution $u$ and the counterfactual solution $u^{cf}_j$.
• Relevance: A term $T_j$ is deemed relevant only if $\delta_j \geq \varepsilon$ for a specified tolerance $\varepsilon$, relative to a specific library, experiment class, and norm.

Theoretical Framework

The paper formalizes six key theoretical results:

1. Residual-Counterfactual Gap Theorem: Proves that the effect of deleting an operator is governed by the inverse linearized PDE map ($L_u^{-1}$), not merely by the residual magnitude. A large residual contribution does not guarantee a large counterfactual deviation if the inverse map dampens the perturbation.
2. Certified Decision Theorem: Establishes error margins for classifying terms as relevant, irrelevant, or abstaining (unresolved) based on the surrogate error bounds of the numerical or neural approximations.
3. Aliasing Theorem: Characterizes non-identifiability through the null space of the operator-evaluation design matrix. If two coefficient vectors are aliased (produce the same design output), no residual-based procedure can distinguish them.
4. Constraint-Manifold Theorem: Proves that operators vanishing on the invariant constraint class of the data (e.g., $\nabla(\nabla \cdot E)$ for divergence-free fields) cannot be identified from trajectories restricted to that class.
5. Pruning-Consistency Theorem: Demonstrates that a two-stage process—sparse residual screening (to propose a candidate support) followed by counterfactual deletion (to prune irrelevant terms)—recovers the functionally relevant support under specific recall and margin conditions.
6. Observable-Level Adjoint Theorem: Extends relevance testing from full-state deviations to specific scientific quantities of interest (e.g., mass, energy, flux) using adjoint sensitivities.

Operational Protocol

The paper outlines a ten-step design protocol (D1–D10) for reusable PDE discovery:

• Anchoring: Using anchored regression to avoid trivial solutions in homogeneous PDEs.
• Design Diagnostics: Reporting coherence and restricted eigenvalue proxies to assess identifiability.
• Conservative Screening: Tuning sparse regression for high recall (accepting false positives) rather than exact support recovery.
• Counterfactual Pruning: Deleting terms from the screened set and computing deviation scores to filter out functionally irrelevant terms.
• Reporting: Explicitly stating the library, experiment class, observation window, norm, and threshold for every relevance claim.

3. Experimental Validation

The framework is validated on both synthetic benchmarks and public geophysical datasets.

Synthetic Experiments

Synthetic cases are designed to verify specific theoretical mechanisms:

• Residual-Counterfactual Gap: Demonstrates cases where a term has high residual contribution but low deletion score.
• Single-Mode Aliasing: Confirms that collinear operator columns (e.g., in a single-mode reaction-diffusion system) prevent separate identification of coefficients.
• Multi-Trajectory Separability: Shows that adding diverse trajectories improves the conditioning of the design matrix and separability of operators.
• Constraint-Null Non-identifiability: Validates that operators vanishing on the constraint class (e.g., $\nabla(\nabla \cdot E)$) yield near-zero deletion scores and are correctly identified as non-identifiable.

Real-Data Experiments

The method is applied to public geophysical fields:

1. Atmospheric Reanalysis (ERA5): Over the Uganda/East Africa window.
   • Result: The design matrix was poorly conditioned (coherence 0.6788, few samples). The sparse screen selected no active operators, and the framework correctly reported the result as unresolved/abstained. This demonstrates the method's ability to avoid overfitting when data is insufficient.
2. NOAA OISST (Sea Surface Temperature): Over the western Indian Ocean.
   • Result: The design was better conditioned. Sparse screening selected five terms ($u_x, u_y, \Delta u, |\nabla u|^2$, constant). Counterfactual pruning reduced this to two functionally relevant terms: the meridional gradient ($u_y$) and the constant term. The other terms, while selected by regression, had deletion scores below the relevance threshold.

4. Key Contributions and Claims

The paper does not claim to automatically recover "true physical laws" from data. Instead, it claims to provide a rigorous diagnostic layer for PDE discovery.

• Distinction of Concepts: It formally separates "residual usefulness" from "counterfactual operator relevance."
• Diagnostic Capability: It provides tools to detect when support recovery is impossible due to aliasing or constraints, rather than treating such failures as optimization errors.
• Screening-to-Pruning Workflow: It establishes a contract where sparse regression acts as a candidate generator, and counterfactual deletion acts as the relevance filter.
• Reporting Standards: It mandates that any discovered operator set be reported as a conditional relevance set relative to a specific experimental design, library, and tolerance, preventing the misinterpretation of local findings as universal physical laws.

5. Significance

The significance of this work lies in its modesty and rigor. It addresses the "black box" nature of current scientific machine learning by providing a theoretical basis for:

1. Why certain terms are selected (or not) beyond simple residual minimization.
2. When a discovery procedure should abstain from making a claim due to insufficient data or design conditioning.
3. How to report results in a way that is reproducible and honest about the limitations imposed by the experiment class and library.

The paper concludes that while it does not solve the problem of unconditional PDE discovery, it offers a necessary framework for distinguishing functional necessity from statistical correlation in data-driven scientific modeling.