Causal prior-embedded physics-informed neural networks and a case study on metformin transport in porous media
Bibliographic record
Abstract
This study introduces a novel approach to transport modelling by integrating experimentally derived causal priors into neural networks. We illustrate this paradigm using a case study of metformin, a ubiquitous pharmaceutical emerging pollutant, and its transport behaviour in sandy media. Specifically, data from metformin's sandy column transport experiment was used to estimate unobservable parameters through a physics-based model Hydrus-1D, followed by a data augmentation to produce a more comprehensive dataset. A causal graph incorporating key variables was constructed, aiding in identifying impactful variables and estimating their causal dynamics or "causal prior." The causal priors extracted from the augmented dataset included underexplored system parameters such as the type-1 sorption fraction F, first-order reaction rate coefficient α, and transport system scale. Their moderate impact on the transport process has been quantitatively evaluated (normalized causal effect 0.0423, -0.1447 and -0.0351, respectively) with adequate confounders considered for the first time. The prior was later embedded into multilayer neural networks via two methods: causal weight initialization and causal prior regularization. Based on the results from AutoML hyperparameter tuning experiments, using two embedding methods simultaneously emerged as a more advantageous practice since our proposed causal weight initialization technique can enhance model stability, particularly when used in conjunction with causal prior regularization. amongst those experiments utilizing both techniques, the R-squared values peaked at 0.881. This study demonstrates a balanced approach between expert knowledge and data-driven methods, providing enhanced interpretability in black-box models such as neural networks for environmental modelling.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".