A Physics-Informed Hybrid Forward-Inverse Machine Learning Framework for High-Dimensional Sparse Recipe Reconstruction

The gist High-dimensional sparse recipe formulation in process industries involves nonlinear association of various covariates, bounded by sparsity constraints and multi-criteria tolerances.

Framework Overview

This study proposes a hybrid physics-informed uncertaintyaware framework structured into three modules: (a) Module A, a multi-target forward quality model; (b) Module B, a sparse inverse reconstructor; and (c) Module C, a feasibility-constrained ranking layer. The methodology is formulated as a general framework for sparse recipe reconstruction and is validated on an industrial textile color matching problem.

Module A: Forward Quality Prediction

Module A functions as a multi-target forward quality model that predicts four shade-quality targets from recipe, process, dye, target reflectance, and spectral features. The forward task is defined as supervised regression from recipe, process, dye, target reflectance, target K/S values, and spectral features to four shade-quality targets expressed as equation 4. The model inputs are expressed as xF i = [ci, pi, si] where pi represents production process parameters and si is the target shade spectral profile comprising reflectance and Kubelka–Munk values across 400–750 nm. The forward model then maps this feature vector to the four shade-quality targets expressed as equation 2.

Module B: Sparse Inverse Recipe Reconstruction

Module B is a sparse inverse reconstructor designed to recover the dye-concentration vector from target shade information using equation 3. The inverse problem is the converse of forward prediction, where the goal is to recover the dye-concentration vector ci from si and pi through equation 3. The inverse input vector is xI i = [si, pi] therefore inverse recipe reconstruction defined as equation 3. Five inverse model families were evaluated for this task, including K-nearest neighbours, Ridge regression, ElasticNet, Random Forest, and Extra Trees. A component-wise median blend achieved a recipe-vector MAE of 0.0295. The sparse post-processing function F(·) converts raw inverse predictions into sparse candidate vectors by clipping negative values to zero and removing components below 0.01% OWF, retaining the three largest predicted components.

Module C: Feasibility-Constrained Selection and Auditing

Module C serves as a feasibility-constrained ranking layer that pools similar or nearly similar historical recipes and uses a forward predictor module to assess the output quality. Historical recipe retrieval was evaluated under three exclusion rules, including self-exclusion, recipe-groupsafe retrieval, and strict unseen-target retrieval. The retrieval score is defined as equation 8. The sparsified inverse-recipes were evaluated by passing them through the fitted forward models to obtain surrogate predictions of ∆ED65, ∆EF11, ∆Emax, and the absolute metamerism gap. Candidate recipes were screened using predefined surrogate acceptance thresholds such as ∆ED65 ≤ 1.0 and absolute metamerism gap ≤ 0.5. The final outputs are a decision support tool that minimizes costly laboratory trials across industrial formulation domains.

Key Findings and Validation

The study validated the framework in an industrial textile dyeing context with recipe-grouped cross-validation under multi-illuminant quality assessment comprising process and production information, target reflectance, K/S values, and spectral descriptors. Histogram Gradient Boosting (HGB) with spectral-only features (Rλ, K/S) and Gradient Boosting Regressor (GBDT) with recipe-process features outperformed other configurations with out-of-Fold MAE of 0.1655 and 0.1523 in the forward quality model. For inverse reconstruction, a component-wise median blend achieved a recipe-vector MAE of 0.0295. Exploratory conditional treatment effect (CATE) suggests an observational association that alkali adjustment reduces error for red and pale shades (CATE = −0.196). The framework provides a human-in-the-loop decision-support tool that minimizes costly laboratory trials across industrial formulation domains.

Conclusion

This study developed and evaluated a physics-informed hybrid machine learning framework for sparse recipe formulation and applies it to textile color reconstruction. The framework can be used as a supporting tool rather than an entirely automatic recipe formulation mechanism due to its current limitation of wet-lab-based validation. Future studies should evaluate the framework using independent production data and controlled dyeing trials.

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The dataset used in this study was collected from an industrial textiledyeing partner in Bangladesh under a non-disclosure agreement for sharing sensitive commercial and industry identifier information. The final feature matrix included recipe-composition, anonymized dye-vector, dye-family, process, categorical fabric, reflectance, K/S, and spectral descriptor features for forward quality prediction and inverse dye-recipe modelling. Recipe records were standardized by normalizing identifiers and dye names. The final dataset contained 212 supervised observations, 85 recipe numbers, and 51 target shades. The total dye concentration defined in the same Sec 3.1. Dye-component entropy quantified as equation 11. Recipe-level dye count, total dye concentration, maximum dye component, dyecomponent entropy, and dye-ratio descriptors were also computed. The entire candidate set contained 222 predictors: 36 dye-concentration variables, 16 dye-composition descriptors, 12 recipe-related variables, 26 process variables, 30 one-hot-encoded categorical variables, and spectral descriptor features. For reflectance Riλ, expressed as a proportion between zero and one, the Kubelka–Munk transformation was calculated as equation (12). The spectral variables covered 400–750 nm at 10 nm intervals. The descriptors included the mean, standard deviation, minimum, maximum, range, area, slope, derivative statistics, and wavelength-band means. Spearman correlation screening (presented in figure 2) and variance inflation factor (VIF) diagnostics were used to assess multicollinearity. The primary forward and inverse models were evaluated using five-fold GroupKFold cross-validation grouped by recipe number. Strict inverse generalization was evaluated using five-fold GroupKFold cross-validation grouped by target identifier so that all observations associated with the same target shade remained within one fold. The 51 target identifiers were distributed across the five validation folds.

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The study also did not directly measure reductions in laboratory trials, development time, or resource use. Inverse models should generate several candidate recipes under chemical, process, concentration, cost, and color-quality constraints. Larger datasets, nested validation, and improved uncertainty methods are also needed before industrial integration.

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This study developed and evaluated a three-module workflow for forward color-quality prediction, sparse inverse reconstruction, and candidate retrieval and screening, together with uncertainty and exploratory interpretation analyses. Different models with different types of configurations were tested and some of them showed their superiority based on different targets. Histogram gradient boosting with spectral-only inputs obtained MAEs of 0.1655 for ∆ED65 and 0.1398 for ∆Emax. Gradient boosting with recipe-process inputs achieved an MAE of 0.1523 for ∆EF11. A quantile random forest achieved an MAE of 0.1046 for the absolute metamerism gap. After multiplicity adjustment, only the selected metamerism-gap model showed a detectable improvement. For inverse recipe reconstruction, a component-wise median ensemble of ten trained base models achieved the lowest recipe-vector MAE of 0.0295. Among the individual models, Extra Trees with spectral inputs achieved a recipe-vector MAE of 0.0338. But inverse performance declined for the unseen target shade, with MAE increasing to 0.0602. There was no optimal model found in this study that could outperform all tasks simultaneously at a time. However, this framework can be used as a supporting tool rather than an entirely automatic recipe formulation mechanism due to its current limitation of wet-lab-based validation. Future studies can explore more robust dataset with wet-lab validation with industrial deployment target.

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The study also did not directly measure reductions in laboratory trials, development time, or resource use. Inverse models should generate several candidate recipes under chemical, process, concentration, cost, and color-quality constraints.

1. textbfCandidate Generation via Hybrid Retrieval and Inverse Modeling: The system can generate sparse recipes by combining historical retrieval with inverse recipe generation. This addresses RQ2 by producing candidates that are grounded in existing industrial data while being constrained by target shade information, as described in Module C's objective of orchestrating the historical retrieval system along with inverse recipe generation to produce a human-in-the-loop decision-support tool.

2. textbfUncertainty Quantification for Quality Screening: The improved system can assess the reliability of predictions by using a physics-informed uncertaintyaware framework in Module A. This allows it to provide 80% prediction intervals, which, when analyzed via prediction-interval coverage and uncertainty informativeness, help technicians understand the confidence before dispensing chemicals, as shown in Table 5 and Figure 12.

3. textbfAdaptive Process Tuning via Causal Inference: The system can suggest optimal process adjustments by utilizing Exploratory Observational Effect and Heterogeneity Analysis. Specifically, it can leverage CATE estimates to guide testing, such as suggesting increasing soda ash for red-family or high-reflectance shades, which translates to actionable insights for the lab technicians in Module C.

4. textbfModel Selection via Multi-Objective Scoring: The system can select the best model by employing a composite scoring function, Sm = MAErecipe,m + 0.1 MAEtotal,m + (1 − Factive1,m), to rank models based on recipe-vector error and active-dye precision. This ensures the chosen forward and inverse models are optimized for practical industrial constraints rather than just raw prediction accuracy.

5. textbfSparse Recipe Post-Processing: The system can enforce hard sparsity constraints by applying a Sparsity post-processing function to convert raw inverse predictions into feasible candidates, specifically by clipping negative values to zero and retaining the three largest predicted components, ensuring the output adheres to industrial constraints like using a maximum of three active dyes (q ≤ 3).