# Excluding the Target Domain Improves Extrapolation: Deconfounded Hierarchical Physics Constraints

arXiv: https://arxiv.org/abs/2605.07485 · HTML: https://mlsanity.com/paper/2605.07485/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 15 of 15)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Machine learning pipeline for battery state of health estimation](https://mlsanity.com/paper/2102.00837/) (2021)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [Dynaformer: A Deep Learning Model for Ageing-aware Battery Discharge Prediction](https://mlsanity.com/paper/2206.02555/) (2022)
- [BatteryLife: A Comprehensive Dataset and Benchmark for Battery Life Prediction](https://mlsanity.com/paper/2502.18807/) (2025)

## Abstract (arXiv)

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).