Excluding the Target Domain Improves Extrapolation: Deconfounded Hierarchical Physics Constraints

Abstract: Extrapolation to out-of-distribution conditions is a fundamental challenge for physics-constrained deep generative models. Existing methods apply physical constraints as a single static regularization term uniformly across the generation process, and address neither the hierarchical structure of physical laws and the confounding variable problem. We propose the Deconfounded Hierarchical Gate (DHG), which serves as a diagnostic and control mechanism: it identifies when and how strongly temperature confounding contaminates each constraint level, so that hierarchical gates reflect intrinsic physical inconsistency rather than spurious temperature effects. DHG combines counterfactual estimation via the do-operator with backdoor adjustment to remove confounding, then applies Coarse-to-Fine physical constraints progressively. We report a counter-intuitive finding in pretraining: excluding the target-domain data from pretraining outperforms including it by 39% in extrapolation performance (RMSE 0.224 vs. 0.324). This occurs because FNO learns domain-agnostic physical patterns that transfer more effectively when the target domain is withheld. On a lithium-ion battery temperature extrapolation benchmark (trained at 24 degrees Celsius, evaluated at 4.0--43.0 degrees Celsius), our method achieves RMSE = 0.215, a 46% improvement over the unconstrained baseline (Pure CFM: 0.397).

• The paper introduces a DHG framework that deconfounds temperature effects and hierarchically schedules physics constraints, leading to improved OOD extrapolation.
• It demonstrates that target-exclusion during pretraining reduces RMSE significantly, with a 46% improvement over standard methods.
• Theoretical contributions include generalization bounds and a principled separation of intrinsic and confounding violations, offering practical guidelines for physics-constrained models.

Excluding the Target Domain in Physics-Constrained Generative Models: Deconfounded Hierarchical Physics Constraints (DHG)

Problem Formulation and Limitations of Prior Approaches

Generalization under distribution shift remains a core bottleneck for physics-constrained deep generative models, particularly when extrapolation to out-of-distribution (OOD) conditions (such as previously unseen temperature regimes) is required. Standard practice in physics-informed modeling incorporates physical constraints as static loss terms applied uniformly across the entire generative process. However, these approaches fail to account for the hierarchical architecture of physical laws and are unable to distinguish intrinsic physical violations from those arising due to confounding factors, notably temperature in electrochemical systems. This oversight inflates model variance and can bias constraint enforcement, impairing OOD generalization.

Proposed Methodology: DHG and Hierarchical Constraint Scheduling

The work introduces the Deconfounded Hierarchical Gate (DHG), an intervention grounded in causal inference using Pearl’s do-operator, which dynamically estimates and removes confounding effects—here, temperature—on constraint satisfaction at multiple levels. This deconfounding is followed by a Coarse-to-Fine scheduling of hierarchical physics constraints, assigning progressively increasing regularization strength from global to local self-consistency as the generative process evolves temporally.

The pipeline proceeds in three stages:

1. Physical Pattern Pretraining: FNO(1), a one-dimensional Fourier Neural Operator pretrained on multi-condition datasets (excluding the target domain), is used to capture condition-invariant physical relationships.
2. Conditional Flow Matching (CFM): The generative model is trained with velocity fields conditioned on the relevant physical context (here, temperature), using the outputs of the frozen FNO(1) as auxiliary physical guidance.
3. Hierarchical Physical Constraint Refinement: DHG quantifies and debiases temperature confounding per-constraint-level and per-generation-timestep via counterfactual estimation and backdoor adjustment, enforcing staged Sigmoid-scheduled constraints.

This framework yields a rigorous diagnostic and control architecture, explicitly distinguishing between violations due to physical inconsistency and those stemming from spurious confounding, and applies constraint gradients only at appropriate points in the generative trajectory.

Principal Empirical Findings

A critical and counter-intuitive empirical finding is the robust superiority of target-exclusion pretraining: FNOs pretrained on a cross-domain dataset omitting the target domain outperform those pretrained with the target included, with a 39% reduction in RMSE for temperature extrapolation (0.224 vs. 0.324). On a challenging extrapolation benchmark (training at $24^\circ$C, evaluated at $4$–$43^\circ$C), the DHG-based method achieves RMSE $= 0.215$, representing a 46% improvement over the unconstrained CFM baseline (RMSE $=0.397$). Fine-tuning the pretrained FNO leads to catastrophic forgetting, confirming the necessity of freezing to preserve invariant physical structure. The diversity of pretraining conditions (rather than loss minimization or reconstruction accuracy) proves to be the determining factor for generalization, upending common assumptions in transfer learning.

Moreover, analysis using DHG demonstrates that the model internalizes and adapts to the confounding structure in the data; learned backdoor coefficients $\beta_{kj}$ saturate in domains with strong confounding signal, and temperature discrimination accuracy of generated waveforms exceeds random by a factor of 2.2 (NASA dataset) and 1.9 (MICH_EXP dataset), quantitatively confirming selective sensitivity to confounding structure.

Theoretical Contributions

The paper provides generalization bounds for hierarchical physics-constrained models, establishing that progressive application of constraints reduces hypothesis class complexity and yields tighter empirical risk bounds. It further rigorously justifies the freezing of the pretrained FNO, linking catastrophic forgetting to a generalization gap when the downstream dataset is significantly smaller than pretraining data. The temperature encoding strategy is shown to convert the extrapolation problem from target temperature to an interpolation problem in embedding space under Arrhenius-structured kinetics, enabling consistency guarantees in the extrapolation domain.

Practical and Theoretical Implications

The findings establish target-exclusion pretraining as a general principle for OOD generalization in physics-constrained models governed by invariant structures, consistent with the invariant risk minimization paradigm. Practically, the approach obviates the need for target domain samples in pretraining, facilitating transfer to unobserved environments. Theoretically, the introduction of DHG enables principled separation between apparent and physical violations, providing robust OOD performance guarantees.

However, the results reveal that while hierarchical physics constraints aid temperature extrapolation, they are ineffective for cycle extrapolation where intrinsic system feedback and non-invariant nonlinear dynamics dominate. This limitation delineates the scope of applicability and motivates explicit modeling of underlying feedback structures for further advances.

Future Directions

Subsequent work may aim to: optimize the DHG clamp bounds for confounding removal; integrate physics-informed validators with explicit degradation features; extend the encoding and constraint scheduling to additional battery chemistries and datasets; and incorporate equivariance in neural operators for enhanced OOD guarantees. Modeling the nonlinear feedback structure for cycle extrapolation remains a critical open direction.

Conclusion

This work substantiates that excluding the target domain in pretraining, together with causal deconfounding and hierarchical coarse-to-fine constraint scheduling, delivers state-of-the-art OOD extrapolation for physics-constrained generative models. The approach not only outperforms baselines across strong distribution shifts but also delivers actionable criteria for evaluating when physical constraints will translate to improved OOD generalization. The integration of causal diagnostics with generative modeling represents a significant step for robust physics-aware AI in scientific applications.