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

Md. Muqtadir Fuad; Md. Istiak Adnan; Ridwan Al Aziz

cs.LGOct 8, 2026

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

Authors: Md. Muqtadir Fuad, Md. Istiak Adnan, Ridwan Al Aziz

Organizations: Department of Industrial and Production Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka 1000, Bangladesh

Abstract

High-dimensional sparse recipe formulation in process industries involves nonlinear association of various covariates, bounded by sparsity constraints and multi-criteria tolerances. Traditional physics-based classical methods (such as Kubelka-Munk) fail to generate sparse candidate recipes properly due to their simplified assumptions and industrial process variations. This study proposes a hybrid physics-informed uncertainty-aware framework that is 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. Then it has been validated 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 ( $R_λ$ , K/S) and Gradient Boosting Regressor (GBDT) with recipe-process features outperforms 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). This framework provides a human-in-the-loop decision-support tool that minimizes costly laboratory trials across industrial formulation domains.

Figures & tables

Symbol	Definition
$i,\ j,\ \lambda$	Indices for observations, dye components, and spectral wavelengths, respectively.
$N,\ D,\ K$	Numbers of observations, active dye variables, and cross-validation folds; $D=36$ and $K=5$ .
$\mathbf{c}_{i}$	Observed dye-recipe concentration vector for observation $i$ .
$c_{ij}$	Concentration of dye $j$ in recipe $i$ , expressed as percentage on weight of fiber.
$T_{i}$	Total dye concentration, $T_{i}=\sum_{j=1}^{D}c_{ij}$ .
$R_{i\lambda}$	Spectral reflectance of observation $i$ at wavelength $\lambda$ .

Table 1: Notations used in this study.

Figure 1: Overview of the data preparation, model development, validation, and decision-support workflow.

Figure 2: Spearman correlation matrix for the selected features and target variables.

Figure 3: Feature preprocessing and multicollinearity-aware feature-selection workflow.

Target	Model	MAE	95% cluster-bootstrap CI	RMSE	Median AE	Bias	Spearman $\boldsymbol{\rho}$
$\Delta E_{\mathrm{D65}}$	HGB- (Sp.)	0.1655	0.1475–0.1853	0.2094	0.1558	0.0069	0.2368
$\Delta E_{\mathrm{F11}}$	GB- (RP)	0.1523	0.1316–0.1759	0.1977	0.1279	0.0040	0.3988
$\Delta E_{\max}$	HGB- (Sp.)	0.1398	0.1230–0.1596	0.1822	0.1228	0.0087	0.2507
Absolute metamerism gap	QRF-LINEAR-KM (LSP)	0.1046	0.0909–0.1191	0.1364	0.0865	-0.0083	0.4746

Table 2: Lowest pooled out-of-fold MAE among the directly evaluated forward and uncertainty models for each color-quality target.

Figure 4: Observed and out-of-fold predicted values for the lowest-MAE model of each color-quality target. Panel A shows

\Delta E_{\mathrm{D65}}

, Panel B shows

\Delta E_{\mathrm{F11}}

, Panel C shows

\Delta E_{\max}

, and Panel D shows the absolute metamerism gap.

Figure 5: Distribution of absolute out-of-fold prediction errors for the lowest-MAE model of each color-quality target.

Figure 6: Lowest pooled out-of-fold MAE by model block and color-quality target.

Figure 7: Relative changes in MAE after input-block removal.

Figure 8: Input-block ablation across models and color-quality targets.

Figure 9: Feature-block permutation importance across models and targets.

Figure 10: Two-way partial-dependence surfaces across models and targets.

Figure 11: Empirical coverage of nominal 80% prediction intervals.

Figure 12: Observed absolute metamerism gaps, median predictions, and nominal 80% prediction intervals for the selected uncertainty method.

Figure 13: Comparison of selected inverse-reconstruction models using recipe-vector MAE, active-dye F1 score, and total-dye absolute error.

Figure 14: Observed and predicted total dye concentrations for the KNN reference, selected Extra Trees base model, component-wise median blend, and strict unseen-target random forest.

Figure 15: Mean best-observed

\Delta E_{\max}

within Top-

k

historical candidate sets under the three retrieval protocols.

Figure 16: Comparison of candidate sources using recipe-vector MAE, total-dye-load absolute error, and active-dye F1. Error bars show 95% recipe-cluster bootstrap confidence intervals.

Inverse model	Candidate form	Forward outcome	Forward model
Median blend	Raw and sparse	$\Delta E_{\mathrm{D65}}$	GB- (TSF)
		$\Delta E_{\mathrm{F11}}$	GB- (TSF)
		$\Delta E_{\max}$	RF- (TSF)
		$\lvert\Delta E_{\mathrm{D65}}-\Delta E_{\mathrm{F11}}\rvert$	RF- (TSF)

Table 8: Inverse-to-forward model mapping used for surrogate screening.

Figure 17: Fold-specific surrogate color-quality predictions for the four candidate sources. Error bars show 95% recipe-cluster bootstrap confidence intervals.

Figure 18: Unit-level CATE distributions (

\hat{\tau}_{i}

) across 8 process and recipe treatments for the 4 quality targets. Negative values indicate error reduction (quality improvement).

Target	High exposure	Subgroup	Mean CATE ( $n$ )
$\Delta E_{\mathrm{D65}}$	Soda ash concentration	Red dye family	$-0.196$ (62)
	Soda ash concentration	High reflectance	$-0.152$ (64)
	Blue-dye ratio	Low reflectance	$+0.132$ (71)
$\Delta E_{\mathrm{F11}}$	Red-dye ratio	Mid reflectance	$+0.159$ (70)
	Red-dye ratio	Blue dye family	$+0.135$ (61)
$\Delta E_{\max}$	Soda ash concentration	Red dye family	$-0.104$ (62)

Table 9: Selected exploratory subgroup CATE estimates.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Model family	Hyperparameter search space	Configurations per target–representation
Random forest	n_estimators $\in\{80,120,200,300,400\}$ ; max_depth $\in\{\texttt{None},3,4,6,8,12\}$ ; min_samples_leaf $\in\{1,2,3,5,8,12\}$ ; min_samples_split $\in\{2,4,8,12,16\}$ ; max_features $\in\{\texttt{sqrt},0.25,0.35,0.50,0.75,1.00\}$ ; bootstrap $\in\{\texttt{True},\texttt{False}\}$ .	200 randomized
Extra Trees	n_estimators $\in\{100,150,250,400,600\}$ ; max_depth $\in\{\texttt{None},3,4,6,8,12\}$ ; min_samples_leaf $\in\{1,2,3,5,8,12\}$ ; min_samples_split $\in\{2,4,8,12,16\}$ ; max_features $\in\{\texttt{sqrt},0.25,0.35,0.50,0.75,1.00\}$ ; bootstrap $\in\{\texttt{False},\texttt{True}\}$ .	200 randomized
Gradient boosting	n_estimators $\in\{50,80,120,180,250,350\}$ ; learning_rate $\in\{0.01,0.02,0.035,0.05,0.08,0.12\}$ ; max_depth $\in\{1,2,3\}$ ; min_samples_leaf $\in\{2,3,5,8,12,16\}$ ; subsample $\in\{0.60,0.75,0.90,1.00\}$ ; max_features $\in\{\texttt{None},\texttt{sqrt},0.50,0.80\}$ .	200 randomized
Histogram gradient boosting	max_iter $\in\{60,80,120,180,250\}$ ; learning_rate $\in\{0.01,0.02,0.035,0.05,0.08,0.12\}$ ; max_leaf_nodes $\in\{5,7,9,15,21,31\}$ ; min_samples_leaf $\in\{5,8,12,16,24,32\}$ ; l2_regularization $\in\{0,0.01,0.05,0.10,0.50,1.00,3.00\}$ ; early_stopping $\in\{\texttt{True},\texttt{False}\}$ .	200 randomized
Ridge regression	$\alpha\in\{0.01,0.03,0.10,0.30,1,3,10,30,100,300\}$ .	10 exhaustive
ElasticNet	$\alpha\in\{0.001,0.003,0.01,0.03,0.10,0.30,1,3,10\}$ ; l1_ratio $\in\{0.05,0.10,0.20,0.35,0.50,0.70,0.90\}$ ; max_iter $\in\{5000,10000,20000\}$ .	189 exhaustive

Appendix

Table A1 : Hyperparameter search spaces used for forward color-quality prediction.

Model family	Hyperparameter search space or sampling rule	Configurations per representation
K-nearest neighbours	n_neighbors $\in\{3,5,7,9,11,15,21\}$ ; weights $\in\{\texttt{uniform},\texttt{distance}\}$ ; Minkowski parameter $p\in\{1,2\}$ .	28 exhaustive
Ridge regression	The fixed values $\alpha\in\{1,3,10\}$ were included. The remaining unique values were sampled as $\alpha=10^{u}$ , where $u\sim\mathcal{U}(-3,2)$ .	200 sampled
ElasticNet	The fixed pairs $(\alpha,l_{1}\text{-ratio})=(0.01,0.10)$ and $(0.03,0.30)$ were included. Remaining values used $\alpha=10^{u}$ , where $u\sim\mathcal{U}(-4,-0.3)$ , and $\texttt{l1\_ratio}\in\{0.05,0.10,0.20,0.40,0.70,0.90\}$ . The maximum number of iterations was fixed at 20,000.	200 sampled
Random forest	n_estimators $\in\{120,200,300,500\}$ ; max_depth $\in\{\texttt{None},4,6,8,12\}$ ; min_samples_leaf $\in\{1,2,3,5,8\}$ ; max_features $\in\{\texttt{sqrt},\texttt{log2},0.35,0.50,0.75,1.00\}$ . Bootstrap sampling was fixed to True .	200 sampled
Extra Trees	n_estimators $\in\{120,200,300,500\}$ ; max_depth $\in\{\texttt{None},4,6,8,12\}$ ; min_samples_leaf $\in\{1,2,3,5,8\}$ ; max_features $\in\{\texttt{sqrt},\texttt{log2},0.35,0.50,0.75,1.00\}$ . Bootstrap sampling was fixed to False .	200 sampled

Appendix

Table A2 : Hyperparameter search spaces used for inverse dye-recipe reconstruction.

Target	Rank	Model	Input representation	MAE	RMSE	Median AE	Bias	Spearman $\boldsymbol{\rho}$
$\Delta E_{\mathrm{D65}}$	1	Histogram gradient boosting	Spectral-only	0.1655	0.2094	0.1558	0.0069	0.2368
	2	Random forest	Spectral-only	0.1657	0.2130	0.1359	0.0039	0.2422
	3	Histogram gradient boosting	Recipe–process	0.1669	0.2121	0.1578	0.0033	0.2455
	4	Gradient boosting	Spectral-only	0.1670	0.2107	0.1448	0.0067	0.2661
	5	Gradient boosting	Tree-safe full	0.1670	0.2124	0.1436	0.0073	0.2224
$\Delta E_{\mathrm{F11}}$	1	Gradient boosting	Recipe–process	0.1523	0.1977	0.1279	0.0040	0.3988

Appendix

Table A3 : Five lowest pooled out-of-fold MAE models for each color-quality target.

Target	Method	Input representation	Median- prediction MAE	95% CI for MAE	Coverage	95% CI for coverage	Mean width	Pinball loss	Width–error Spearman $\boldsymbol{\rho}$	Coverage compatible	Selected
$\Delta E_{\mathrm{D65}}$	Fold-wise empirical quantiles	Response only	0.1733	0.1546–0.1933	0.7877	0.7277–0.8436	0.5520	0.0517	0.0512	Yes	Yes
	Quantile random forest	Linear-safe plus K/S	0.1762	0.1585–0.1960	0.4387	0.3689–0.5139	0.2836	0.0611	$-0.0228$	No	No
	Quantile random forest	Tree-safe plus K/S	0.1749	0.1559–0.1962	0.4670	0.3981–0.5415	0.3043	0.0599	$-0.0322$	No	No
$\Delta E_{\mathrm{F11}}$	Fold-wise empirical quantiles	Response only	0.1676	0.1451–0.1925	0.7925	0.7297–0.8512	0.4971	0.0522	$-0.0600$	Yes	Yes
	Quantile random forest	Linear-safe plus K/S	0.1599	0.1361–0.1859	0.5283	0.4465–0.6073	0.2580	0.0571	0.0386	No	No
	Quantile random forest	Tree-safe plus K/S	0.1572	0.1366–0.1797	0.4151	0.3333–0.4951	0.2124	0.0589	0.1300	No	No

Appendix

Table A4 : Complete comparison of uncertainty methods for the four color-quality targets.

Dye	Observed active	Predicted active	MAE	RMSE	Bias	Precision	Recall	F1
Dye A	65	69	0.12570	0.29253	0.01670	0.8406	0.8923	0.8657
Dye B	70	87	0.24896	0.63326	0.10403	0.6782	0.8429	0.7516
Dye C	42	39	0.02160	0.08548	-0.00903	1.0000	0.9286	0.9630
Dye D	0	0	0.00000	0.00000	0.00000	0.0000	0.0000	0.0000
Dye E	54	49	0.10598	0.28474	-0.01246	0.7551	0.6852	0.7184
Dye F	31	30	0.11577	0.43177	-0.00616	0.6667	0.6452	0.6557

Appendix

Table A8 : Dye-specific reconstruction results for the selected target-spectral Extra Trees model.

Figure 19: Dye-specific concentration MAE for the 15 most difficult components under the selected grouped-validation inverse model.

Figure 20: Recipe-vector mean absolute error for the inverse base-model configurations under recipe-grouped cross-validation.

Figure 21: Active-dye F1 scores for the inverse base-model configurations under recipe-grouped cross-validation.

Author (Year)	Process	Variables	Dataset	Dataset Size	Country
(Zhu et al., 2025)	Full-color-gamut rotor spinning & Neural Network color prediction	Blending ratios (8 primary fibers), spectral reflectance, CIELAB	Blended yarn fabrics (experimental samples)	273 samples (255 training, 18 validation)	China
(Cui et al., 2025)	Bidirectional color prediction using BPNN models & Grid-based color mixing	Inputs/Outputs: Spectral reflectance, Blend ratios (4 primary colors), CIELAB values	Experimental color-spun yarn samples	66 distinct yarn samples	China
(Xie et al., 2024)	GBRT (Gradient-Boosting Regression Tree) with Log-transformation	Dye concentrations (3 disperse dyes), full reflectance spectra, CIELAB/RGB	Industrial dyeing records (Polyester fabrics)	356 industrial dyeing data	China
(Souissi et al., 2024)	Genetic Algorithm (GA) optimization (Roulette wheel selection)	Disperse dye concentrations (3 dyes), CMC color difference	Dyed PET/PTT bicomponent filaments	15 reference colors	Tunisia
(Zhou et al., 2024)	NSGA-II (Genetic Algorithm) + ESM-EWM-TOPSIS decision model	Dye concentrations, CMC $\Delta E$ , metamerism index, cost	Laboratory dyeing dataset (Cotton/Reactive dyes)	40 target colors	China
(Sun et al., 2023)	Neuronal Network (BPNN) + Gridded color-mixing model	Inputs/Outputs: Fiber mixing ratios (4 colors) and Spectral reflectance	Gridded color-mixed yarn samples	41 samples (31 Training, 10 Validation)	China

Appendix

Table A9 : Different machine-learning architectures and optimization techniques to solve the non-linear dye recipe formulation and color-matching problem

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