Symbolic Machine Learning for Vapor-Liquid Equilibrium Prediction in Cx-N2 Binary Mixtures
Summary
This paper proposes a symbolic machine learning approach to discover interpretable corrections to the Peng-Robinson equation of state for predicting vapor-liquid equilibrium in hydrocarbon-nitrogen binary mixtures, improving accuracy over the original EOS.
arXiv:2608.11255v1 Announce Type: new
Abstract: Accurate prediction of vapor--liquid equilibrium (VLE) for hydrocarbon-nitrogen mixtures remains challenging for cubic equations of state, particularly across broad ranges of composition and hydrocarbon chain length. While deep learning models can provide accurate predictions, they often lack interpretability and explicit analytical expressions. In this work, we propose a symbolic machine learning approach to discover interpretable symbolic corrections to Peng-Robinson equation-of-state (PR-EOS) predictions from experimental data. The proposed approach adopts a two-level strategy: symbolic expressions are first identified for individual hydrocarbon systems, after which their coefficients are represented as functions of carbon number to enable accurate prediction across different hydrocarbon systems. The results demonstrate significantly improved prediction accuracy over the original PR-EOS across all hydrocarbon-nitrogen systems. Overall, the proposed approach provides an interpretable symbolic correction framework for improving PR-EOS predictions of hydrocarbon-nitrogen VLE.
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# Symbolic Machine Learning for Vapor–Liquid Equilibrium Prediction in 𝐶_𝑥–𝑁₂ Binary Mixtures Source: [https://arxiv.org/html/2608.11255](https://arxiv.org/html/2608.11255) Suman ChakrabortyGary HuangMehek MathurGuang LinLi Qiao[lqiao@purdue\.edu](mailto:[email protected]) ###### Abstract Accurate prediction of vapor–liquid equilibrium \(VLE\) for hydrocarbon–nitrogen mixtures remains challenging for cubic equations of state, particularly across broad ranges of composition and hydrocarbon chain length\. While deep learning models can provide accurate predictions, they often lack interpretability and explicit analytical expressions\. In this work, we propose a symbolic machine learning approach to discover interpretable symbolic corrections to Peng–Robinson equation\-of\-state \(PR\-EOS\) predictions from experimental data\. The proposed approach adopts a two\-level strategy: symbolic expressions are first identified for individual hydrocarbon systems, after which their coefficients are represented as functions of carbon number to enable accurate prediction across different hydrocarbon systems\. The results demonstrate significantly improved prediction accuracy over the original PR\-EOS across all hydrocarbon–nitrogen systems\. Overall, the proposed approach provides an interpretable symbolic correction framework for improving PR\-EOS predictions of hydrocarbon–nitrogen VLE\. ###### keywords: Peng\-Robinson Equation of State , Phase Equilibrium , Binary Mixtures , Symbolic Regression ††journal:arXivPDEpartial differential equationFEfinite elementFEM[finite element](https://arxiv.org/html/2608.11255#id4.2.id2)\([FE](https://arxiv.org/html/2608.11255#id4.2.id2)\) methodRBreduced basisCRBcertified[reduced basis](https://arxiv.org/html/2608.11255#id6.4.id4)\([RB](https://arxiv.org/html/2608.11255#id6.4.id4)\)EIMempirical interpolation methodSCMsuccessive constraint methodSCRBEstatic condensation reduced basis elementSCstatic condensationPRport reductionPR\-SCRBE[port reduction](https://arxiv.org/html/2608.11255#id12.10.id10)\([PR](https://arxiv.org/html/2608.11255#id12.10.id10)\)\-[static condensation reduced basis element](https://arxiv.org/html/2608.11255#id10.8.id8)\([SCRBE](https://arxiv.org/html/2608.11255#id10.8.id8)\)PR\-RBport\-reduced[RB](https://arxiv.org/html/2608.11255#id6.4.id4)DOFdegrees of freedomMLmachine learningPODproper orthogonal decompositionROMreduced\-order modellingRSMresponse surface methodologyCADcomputer\-aided designMAmanifold\-alignmentCRMcommon research modelRSMresponse surface methodologyNNneural networkRBFradial basis functionMAmanifold\-alignmentACarchetype componentICinstantiated componentMDOmultidisciplinary design optimizationMUMPSMUltifrontal Massively Parallel sparse direct SolverGPRGaussian process regressionGPGaussian processQoIsquantities of interest \\affiliation \[label1\]organization=Purdue University, School of Mechanical Engineering,postcode=47906, state=IN, country=United States \\affiliation \[label2\]organization=Purdue University, Department of Mathematics, postcode=47906, state=IN, country=United States \\affiliation \[label3\]organization=Purdue University, School of Aeronautics and Astronautics, postcode=47906, state=IN, country=United States ## 1Introduction Accurate thermodynamic modeling of multicomponent hydrocarbon mixtures is essential in many engineering applications, including high\-pressure fuel injection\[[30](https://arxiv.org/html/2608.11255#bib.bib3)\]and supercritical combustion\[[32](https://arxiv.org/html/2608.11255#bib.bib1),[39](https://arxiv.org/html/2608.11255#bib.bib32)\]\. Reliable vapor–liquid equilibrium \(VLE\) prediction is particularly important because phase behavior strongly influences fuel atomization, transport processes, and thermodynamic efficiency\[[31](https://arxiv.org/html/2608.11255#bib.bib2),[29](https://arxiv.org/html/2608.11255#bib.bib4)\]\. To predict VLE, equations of state \(EOS\) have become one of the most widely used thermodynamic models for describing pressure–temperature–composition relationships in multicomponent mixtures\. Among cubic equations of state, the Peng–Robinson \(PR\)\-EOS\[[23](https://arxiv.org/html/2608.11255#bib.bib18)\]has become one of the most widely used thermodynamic models for predicting vapor–liquid equilibria as well as volumetric and thermodynamic properties of pure substances and mixtures\. Extensive efforts have been devoted to improving the PR\-EOS through modifications of the attraction term, volume translation, additional model terms, and mixing rules\[[18](https://arxiv.org/html/2608.11255#bib.bib19)\]\. Despite its widespread use, the classical PR\-EOS still exhibits non\-negligible prediction errors for hydrocarbon mixtures, particularly under high\-pressure and near\-critical conditions where complex intermolecular interactions become increasingly important\[[18](https://arxiv.org/html/2608.11255#bib.bib19),[28](https://arxiv.org/html/2608.11255#bib.bib33)\]\. Considerable effort has therefore been devoted to improving the predictive capability of the PR\-EOS\. One important direction is the development of predictive group\-contribution mixing rules for hydrocarbon mixtures\[[26](https://arxiv.org/html/2608.11255#bib.bib8),[27](https://arxiv.org/html/2608.11255#bib.bib9)\]\. Other studies have proposed generalized correlations for estimating binary interaction parameters over broad thermodynamic conditions\[[8](https://arxiv.org/html/2608.11255#bib.bib20),[19](https://arxiv.org/html/2608.11255#bib.bib21),[2](https://arxiv.org/html/2608.11255#bib.bib22)\]\. In parallel, extensive experimental measurements have been reported for hydrocarbon–nitrogen systems\. Representative VLE datasets include nitrogen–decane\[[12](https://arxiv.org/html/2608.11255#bib.bib6)\], nitrogen–dodecane\[[11](https://arxiv.org/html/2608.11255#bib.bib17)\], and nitrogen–hydrocarbon systems at elevated pressures\[[3](https://arxiv.org/html/2608.11255#bib.bib10),[17](https://arxiv.org/html/2608.11255#bib.bib11)\]\. High\-pressure solubility measurements have also been reported for heavy n\-alkanes and other hydrocarbon systems\[[6](https://arxiv.org/html/2608.11255#bib.bib13),[25](https://arxiv.org/html/2608.11255#bib.bib14),[22](https://arxiv.org/html/2608.11255#bib.bib15),[38](https://arxiv.org/html/2608.11255#bib.bib12),[10](https://arxiv.org/html/2608.11255#bib.bib16)\]\. These experimental datasets provide valuable benchmarks for validating thermodynamic models\. Recently, machine learning has emerged as an alternative framework for thermodynamic modeling\. Artificial neural networks have been applied to vapor–liquid equilibrium \(VLE\) prediction\[[24](https://arxiv.org/html/2608.11255#bib.bib24),[35](https://arxiv.org/html/2608.11255#bib.bib25)\]\. Subsequent studies extended neural\-network\-based VLE prediction to a wider range of binary systems, including refrigerant mixtures\[[20](https://arxiv.org/html/2608.11255#bib.bib23),[21](https://arxiv.org/html/2608.11255#bib.bib27),[9](https://arxiv.org/html/2608.11255#bib.bib26)\]\. Neural networks have also been employed for hydrocarbon mixtures, equilibrium K\-value prediction, and asymmetric binary systems\[[15](https://arxiv.org/html/2608.11255#bib.bib29),[14](https://arxiv.org/html/2608.11255#bib.bib28),[1](https://arxiv.org/html/2608.11255#bib.bib30),[4](https://arxiv.org/html/2608.11255#bib.bib36)\]\. While these methods generally improve prediction accuracy, fully data\-driven neural\-network models do not provide explicit analytical expressions, making the learned thermodynamic relationships difficult to interpret\. To address this limitation, symbolic regression has emerged as an attractive alternative for discovering explicit mathematical expressions directly from data\[[16](https://arxiv.org/html/2608.11255#bib.bib34),[34](https://arxiv.org/html/2608.11255#bib.bib35)\]\. Instead of fitting a predefined functional form, symbolic regression simultaneously identifies both the analytical structure and the associated numerical coefficients of the underlying relationships\. Consequently, the resulting models remain compact and interpretable while retaining sufficient flexibility to capture nonlinear thermodynamic behavior\. These characteristics make symbolic regression particularly suitable for constructing interpretable corrections to existing physics\-based models rather than replacing them entirely\. A recent study demonstrated the potential of symbolic regression for improving cubic equations of state through data\-driven modifications of EOS parameters for liquid\-phase density prediction\[[40](https://arxiv.org/html/2608.11255#bib.bib43)\]\. In contrast, the present study considers symbolic regression for vapor–liquid equilibrium prediction in multicomponent hydrocarbon–nitrogen mixtures\. In this work, we propose a symbolic machine learning framework for constructing interpretable post\-processing corrections to the PR\-EOS for hydrocarbon–nitrogen vapor–liquid equilibrium prediction\. Rather than directly replacing the underlying thermodynamic model, the proposed approach learns symbolic correction functions for the residual errors of the PR\-EOS, thereby preserving its physical foundation while systematically reducing prediction errors\. The framework adopts a two\-level learning strategy\. First, system\-specific symbolic expressions are independently discovered for each hydrocarbon system, allowing the dominant correction mechanisms to be identified without imposing predefined functional forms\. Next, recurring symbolic structures shared across different hydrocarbon systems are extracted to construct a common symbolic basis\. Finally, only the basis coefficients are parameterized as smooth functions of the carbon number, yielding a unified symbolic correction model that generalizes across multiple hydrocarbon systems while preserving compact analytical expressions and physical interpretability\. The main contributions of this work are summarized as follows: 1. 1\.Interpretable symbolic correction of the PR\-EOS\.The proposed approach learns explicit symbolic correction functions for the residual errors of the PR\-EOS, preserving the underlying thermodynamic model while improving prediction accuracy\. 2. 2\.Two\-level symbolic machine learning\.System\-specific symbolic expressions are first discovered independently, after which shared symbolic basis functions and carbon\-number\-dependent coefficients are identified to construct a unified symbolic correction model\. 3. 3\.Accurate symbolic VLE prediction\.The proposed approach consistently improves pressure and vapor\-phase composition predictions while maintaining compact closed\-form symbolic expressions\. The remainder of this paper is organized as follows\. Section 2 presents the proposed multilevel symbolic regression method\. Section 3 describes the experimental datasets and presents the numerical results\. Finally, Section 4 concludes the paper and discusses future research directions\. ## 2Peng–Robinson Equation of State \(PR\-EOS\) We begin by reviewing the PR\-EOS\[[18](https://arxiv.org/html/2608.11255#bib.bib19),[4](https://arxiv.org/html/2608.11255#bib.bib36)\], which serves as the baseline thermodynamic model throughout this work\. The proposed symbolic machine learning approach is designed to correct the prediction errors of the PR\-EOS rather than replace the underlying thermodynamic formulation\. This section therefore briefly summarizes the governing equations of the PR\-EOS and the corresponding phase\-equilibrium calculations used throughout the remainder of this paper\. For a single\-phase mixture, the pressure is expressed as P=RTv−b−a\(T\)v\(v\+b\)\+b\(v−b\)\.P=\\frac\{RT\}\{v\-b\}\-\\frac\{a\(T\)\}\{v\(v\+b\)\+b\(v\-b\)\}\.\(1\) Here,RRis the universal gas constant,TTis the absolute temperature, andvvis the molar volume\. The parametera\(T\)a\(T\)represents the attractive intermolecular interactions, whereasbbaccounts for the excluded\-volume effect arising from the finite molecular size\. These parameters are determined from the critical properties and acentric factors of the individual components\. For each componentii, the attractive parameteraia\_\{i\}and the covolume parameterbib\_\{i\}are evaluated as ai\\displaystyle a\_\{i\}=0\.45724R2Tc,i2Pc,iαi\(T\),\\displaystyle=0\.45724\\,\\frac\{R^\{2\}T\_\{c,i\}^\{2\}\}\{P\_\{c,i\}\}\\,\\alpha\_\{i\}\(T\),\(2\)bi\\displaystyle b\_\{i\}=0\.07780RTc,iPc,i\.\\displaystyle=0\.07780\\,\\frac\{RT\_\{c,i\}\}\{P\_\{c,i\}\}\.\(3\) The temperature\-dependent attraction function is αi\(T\)=\[1\+\(0\.37464\+1\.54226ωi−0\.26992ωi2\)\(1−T/Tc,i\)\]2,\\alpha\_\{i\}\(T\)=\\left\[1\+\\left\(0\.37464\+1\.54226\\omega\_\{i\}\-0\.26992\\omega\_\{i\}^\{2\}\\right\)\\left\(1\-\\sqrt\{T/T\_\{c,i\}\}\\right\)\\right\]^\{2\},\(4\)whereTc,iT\_\{c,i\},Pc,iP\_\{c,i\}, andωi\\omega\_\{i\}denote the critical temperature, critical pressure, and acentric factor of componentii, respectively\. For a multicomponent mixture with mole fractionsxix\_\{i\}, the pure\-component parameters are combined through the conventional quadratic mixing rules, a\\displaystyle a=∑i∑jxixjaij,\\displaystyle=\\sum\_\{i\}\\sum\_\{j\}x\_\{i\}x\_\{j\}a\_\{ij\},\(5\)b\\displaystyle b=∑ixibi,\\displaystyle=\\sum\_\{i\}x\_\{i\}b\_\{i\},\(6\)where aij=\(1−kij\)aiaj\.a\_\{ij\}=\(1\-k\_\{ij\}\)\\sqrt\{a\_\{i\}a\_\{j\}\}\.\(7\) Here,kijk\_\{ij\}denotes the binary interaction parameter, which accounts for non\-ideal interactions between unlike species\. It satisfies the symmetry conditionkij=kjik\_\{ij\}=k\_\{ji\}together withkii=0k\_\{ii\}=0\. Introducing the dimensionless parameters A=aPR2T2,B=bPRT,Z=PvRT,A=\\frac\{aP\}\{R^\{2\}T^\{2\}\},\\qquad B=\\frac\{bP\}\{RT\},\\qquad Z=\\frac\{Pv\}\{RT\},\(8\)the compressibility factorZZis obtained by solving the cubic equation Z3−\(1−B\)Z2\+\(A−3B2−2B\)Z−\(AB−B2−B3\)=0\.Z^\{3\}\-\(1\-B\)Z^\{2\}\+\(A\-3B^\{2\}\-2B\)Z\-\(AB\-B^\{2\}\-B^\{3\}\)=0\.\(9\) The physically appropriate real root is selected according to the phase of interest\. Phase equilibrium calculations are subsequently performed through fugacity equality between the liquid and vapor phases\. Within the PR\-EOS, the fugacity coefficient of componentiiis evaluated as lnϕi=\\displaystyle\\ln\\phi\_\{i\}=\{\}bib\(Z−1\)−ln\(Z−B\)\\displaystyle\\frac\{b\_\{i\}\}\{b\}\(Z\-1\)\-\\ln\(Z\-B\)−A22B\(2∑jxjaija−bib\)ln\(Z\+\(1\+2\)BZ\+\(1−2\)B\)\.\\displaystyle\-\\frac\{A\}\{2\\sqrt\{2\}B\}\\left\(\\frac\{2\\sum\_\{j\}x\_\{j\}a\_\{ij\}\}\{a\}\-\\frac\{b\_\{i\}\}\{b\}\\right\)\\ln\\left\(\\frac\{Z\+\(1\+\\sqrt\{2\}\)B\}\{Z\+\(1\-\\sqrt\{2\}\)B\}\\right\)\.\(10\) The fugacity coefficients provide the thermodynamic equilibrium criterion for each component and are therefore the key quantities for vapor–liquid equilibrium calculations\. In practical VLE calculations, the computational procedure consists of the following steps: 1. 1\.Evaluate the pure\-component parametersaia\_\{i\},bib\_\{i\}, andαi\(T\)\\alpha\_\{i\}\(T\)\. 2. 2\.Construct the mixture parametersaaandbbusing the quadratic mixing rules\. 3. 3\.Solve the cubic equation of state to obtain the compressibility factorZZ\. 4. 4\.Compute the fugacity coefficients for each component in the liquid and vapor phases\. 5. 5\.Determine the equilibrium pressure and vapor composition by satisfying the phase\-equilibrium conditions\. ## 3Symbolic regression for correction of PR\-EOS ### 3\.1System\-specific symbolic regression Symbolic regression aims to discover explicit mathematical expressions directly from data by simultaneously identifying both the analytical structure and the associated numerical coefficients\. In this work, symbolic regression is performed using the evolutionary search algorithm implemented in PySR\[[5](https://arxiv.org/html/2608.11255#bib.bib37)\], as illustrated in Fig\.[1](https://arxiv.org/html/2608.11255#S3.F1)\. During each generation, candidate symbolic expressions are modified through three complementary genetic operators\. First, mutation introduces structural diversity by replacing a randomly selected subtree with a newly generated symbolic subtree, thereby exploring new functional forms while preserving the remaining expression structure, as shown in Fig\.[1](https://arxiv.org/html/2608.11255#S3.F1)a\. Second, crossover combines information from two parent expressions by exchanging randomly selected subtrees, enabling the construction of offspring that inherit advantageous components from both parents, as shown in Fig\.[1](https://arxiv.org/html/2608.11255#S3.F1)b\. Third, migration periodically transfers high\-quality expressions between independent populations, promoting information sharing while maintaining diversity across parallel evolutionary searches, as shown in Fig\.[1](https://arxiv.org/html/2608.11255#S3.F1)c\. After these structural modifications, each candidate expression is algebraically simplified and its numerical constants are refined through numerical optimization before its fitness is evaluated\. This iterative process jointly explores the symbolic search space while progressively improving the predictive accuracy of the discovered expressions\. Figure 1:Illustration of the evolutionary search process in symbolic regression\. \(a\) Mutation replaces a randomly selected subtree with a newly generated symbolic subtree, modifying the expression fromexp\(0\.63\+x2\)\\exp\(0\.63\+x\_\{2\}\)toexp\(x1tanh\(x2\)\+x2\)\\exp\(x\_\{1\}\\tanh\(x\_\{2\}\)\+x\_\{2\}\)\. \(b\) Crossover exchanges randomly selected subtrees between two parent expressions to generate new candidate expressions\. \(c\) Migration transfers symbolic expressions between independent populations during parallel evolution\. After structural updates, symbolic simplification and numerical constant optimization are subsequently applied before evaluating the updated expressions\.Table 1:Summary of the experimental high\-pressure VLE datasets for binary N2\+ n\-alkane systems used in this study\.SystemSourceTemperature levels \(K\)PmaxP\_\{\\max\}\(MPa\)Data pointsN2\+ n\-C5H12\[[36](https://arxiv.org/html/2608.11255#bib.bib39)\]344\.3, 377\.9, 411\.0,427\.1, 447\.93571N2\+ n\-C6H14\[[7](https://arxiv.org/html/2608.11255#bib.bib40)\]344\.6, 378\.2, 411\.1,444\.5, 466\.0, 488\.45072N2\+ n\-C7H16\[[13](https://arxiv.org/html/2608.11255#bib.bib41)\]313\.6, 344\.3, 377\.8, 411\.0,444\.5, 466\.0, 488\.2,502\.7, 513\.1, 523\.750136N2\+ n\-C9H20\[[37](https://arxiv.org/html/2608.11255#bib.bib42)\]344\.3, 411\.1, 466\.0,502\.7, 543\.45070N2\+ n\-C10H22\[[12](https://arxiv.org/html/2608.11255#bib.bib6)\]344\.3, 377\.9, 411\.0, 444\.5,488\.2, 523\.7, 563\.150142N2\+ n\-C12H26\[[11](https://arxiv.org/html/2608.11255#bib.bib17)\]344\.4, 377\.9, 411\.1, 444\.6,488\.2, 523\.7, 593\.560169The experimental datasets employed in this study are summarized in Table[1](https://arxiv.org/html/2608.11255#S3.T1)\. The datasets consist of high\-pressure VLE measurements for six binary N2\+ n\-alkane systems spanning n\-pentane to n\-dodecane\. For each system, measurements were conducted over multiple temperature levels covering a wide range of thermodynamic conditions, providing a total of 660 experimental data points for model development and evaluation\. For each hydrocarbon–nitrogen system, the experimental VLE measurements are used directly to construct the symbolic regression dataset\. The reduced temperatureTrT\_\{r\}and liquid\-phase nitrogen mole fractionxN2x\_\{\\mathrm\{N\}\_\{2\}\}are taken as input variables, while the regression targets are defined as the residuals between the experimental measurements and the corresponding PR\-EOS predictions\. Specifically, ΔP\\displaystyle\\Delta P=Pexp−PPR,\\displaystyle=P\_\{\\mathrm\{exp\}\}\-P\_\{\\mathrm\{PR\}\},\(11\)ΔyN2\\displaystyle\\Delta y\_\{\\mathrm\{N\}\_\{2\}\}=yexp−yPR,\\displaystyle=y\_\{\\mathrm\{exp\}\}\-y\_\{\\mathrm\{PR\}\},\(12\)wherePexpP\_\{\\mathrm\{exp\}\}andyexpy\_\{\\mathrm\{exp\}\}denote the measured equilibrium pressure and vapor\-phase nitrogen mole fraction, respectively, andPPRP\_\{\\mathrm\{PR\}\}andyPRy\_\{\\mathrm\{PR\}\}are the corresponding PR\-EOS predictions\. Separate symbolic regression models are then identified for the pressure and vapor\-composition corrections in each binary system\. Algorithm[1](https://arxiv.org/html/2608.11255#algorithm1)summarizes the proposed system\-specific symbolic learning procedure\. Rather than directly learning the equilibrium pressure and vapor composition, symbolic regression is applied to the residual errors of the PR\-EOS, allowing the underlying thermodynamic formulation to be retained while correcting its systematic prediction errors\. For each hydrocarbon–nitrogen system, separate symbolic regression models are constructed for the pressure and vapor\-phase composition residuals because these quantities exhibit distinct nonlinear behaviors\. The symbolic expressions are evolved using the procedure described in the previous section, and the final expressions are selected from the Pareto front by balancing prediction accuracy and symbolic complexity\. The learned correction functions are then added to the original PR\-EOS predictions to obtain the corrected pressure and vapor composition\. Require :System data 𝒟\(s\)=\{\(Tri,xN2i,PPRi,yPRi,Pexpi,yexpi\)\}i=1Ns\\mathcal\{D\}^\{\(s\)\}=\\\{\(T\_\{r\}^\{i\},x\_\{N\_\{2\}\}^\{i\},P\_\{\\mathrm\{PR\}\}^\{i\},y\_\{\\mathrm\{PR\}\}^\{i\},P\_\{\\mathrm\{exp\}\}^\{i\},y\_\{\\mathrm\{exp\}\}^\{i\}\)\\\}\_\{i=1\}^\{N\_\{s\}\}\. Ensure :Corrected predictions Pcorr\(s\)\(Tr,xN2\)P\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)and ycorr\(s\)\(Tr,xN2\)y\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)\. ⊳\\trianglerightResidual construction and normalization Construct ΔPi=Pexpi−PPRi\\Delta P^\{i\}=P\_\{\\mathrm\{exp\}\}^\{i\}\-P\_\{\\mathrm\{PR\}\}^\{i\}and Δyi=yexpi−yPRi\\Delta y^\{i\}=y\_\{\\mathrm\{exp\}\}^\{i\}\-y\_\{\\mathrm\{PR\}\}^\{i\}for all samples Normalize \(Tr,xN2\)\(T\_\{r\},x\_\{N\_\{2\}\}\), ΔP\\Delta P, and Δy\\Delta ywithin system ss ⊳\\trianglerightIndependent symbolic regression Perform symbolic regression for \(Tr,xN2\)↦ΔP\(T\_\{r\},x\_\{N\_\{2\}\}\)\\mapsto\\Delta P Perform symbolic regression for \(Tr,xN2\)↦Δy\(T\_\{r\},x\_\{N\_\{2\}\}\)\\mapsto\\Delta y Select the final symbolic expressions from the Pareto front according to prediction accuracy and symbolic complexity ⊳\\trianglerightPR\-EOS correction Construct Pcorr\(s\)=PPR\+ΔP\(s\)\(Tr,xN2\)P\_\{\\mathrm\{corr\}\}^\{\(s\)\}=P\_\{\\mathrm\{PR\}\}\+\\Delta P^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)and ycorr\(s\)=yPR\+Δy\(s\)\(Tr,xN2\)y\_\{\\mathrm\{corr\}\}^\{\(s\)\}=y\_\{\\mathrm\{PR\}\}\+\\Delta y^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\) return*Pcorr\(s\)\(Tr,xN2\)P\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)andycorr\(s\)\(Tr,xN2\)y\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)* Algorithm 1System\-specific symbolic regression for PR\-EOS correctionAlgorithm[2](https://arxiv.org/html/2608.11255#algorithm2)summarizes the proposed multilevel symbolic regression procedure\. Rather than maintaining an independent symbolic expression for each hydrocarbon–nitrogen system, the proposed approach constructs a unified symbolic correction model by identifying symbolic structures shared across multiple systems\. To this end, the system\-specific symbolic expressions obtained in the first stage are first analyzed to extract their constituent symbolic basis functions\. Recurring symbolic structures are subsequently identified to construct shared symbolic basis libraries for the pressure and vapor\-composition corrections\. For each hydrocarbon system, the correction function is represented as a linear combination of the shared symbolic basis functions, ΔP\(s\)\(Tr,xN2\)=∑m=15αm\(P,s\)ϕm\(P\)\(Tr,xN2\),\\Delta P^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)=\\sum\_\{m=1\}^\{5\}\\alpha\_\{m\}^\{\(P,s\)\}\\phi\_\{m\}^\{\(P\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\),\(13\)whereϕm\(P\)\\phi\_\{m\}^\{\(P\)\}denotes themmth shared symbolic basis function andαm\(P,s\)\\alpha\_\{m\}^\{\(P,s\)\}is its corresponding coefficient for systemss\. Likewise, the vapor\-composition correction is represented as Δy\(s\)\(Tr,xN2\)=∑m=16αm\(y,s\)ϕm\(y\)\(Tr,xN2\),\\Delta y^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)=\\sum\_\{m=1\}^\{6\}\\alpha\_\{m\}^\{\(y,s\)\}\\phi\_\{m\}^\{\(y\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\),\(14\) Rather than estimating these coefficients independently for each hydrocarbon system, the proposed approach parameterizes them as functions of the carbon number, allowing the symbolic basis functions to remain common while their coefficients vary systematically with hydrocarbon chain length\. The basis\-internal constants together with the coefficient functions are then optimized over the merged dataset using gradient\-based optimization\[[33](https://arxiv.org/html/2608.11255#bib.bib38)\]\. Finally, the learned symbolic corrections are added to the original PR\-EOS predictions to obtain Pcorr\(s\)\\displaystyle P\_\{\\mathrm\{corr\}\}^\{\(s\)\}=PPR\+ΔP\(s\)\(Tr,xN2\),\\displaystyle=P\_\{\\mathrm\{PR\}\}\+\\Delta P^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\),\(15\)ycorr\(s\)\\displaystyle y\_\{\\mathrm\{corr\}\}^\{\(s\)\}=yPR\+Δy\(s\)\(Tr,xN2\),\\displaystyle=y\_\{\\mathrm\{PR\}\}\+\\Delta y^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\),yielding the final predictionsPcorr\(Tr,xN2,nC\)P\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)andycorr\(Tr,xN2,nC\)y\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)\. Require :Merged dataset 𝒟=⋃s𝒟\(s\)\\mathcal\{D\}=\\bigcup\_\{s\}\\mathcal\{D\}^\{\(s\)\}with system descriptor nCn\_\{C\}\. Ensure :Corrected predictions Pcorr\(Tr,xN2,nC\)P\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)and ycorr\(Tr,xN2,nC\)y\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)\. ⊳\\trianglerightShared symbolic basis construction Extract symbolic basis functions from the system\-specific expressions for ΔP\\Delta P Extract symbolic basis functions from the system\-specific expressions for Δy\\Delta y Construct the shared basis library \{ϕm\(P\)\}m=1MP\\\{\\phi\_\{m\}^\{\(P\)\}\\\}\_\{m=1\}^\{M\_\{P\}\}for ΔP\\Delta P Construct the shared basis library \{ϕm\(y\)\}m=1My\\\{\\phi\_\{m\}^\{\(y\)\}\\\}\_\{m=1\}^\{M\_\{y\}\}for Δy\\Delta y ⊳\\trianglerightIndependent coefficient learning Represent ΔP=∑m=1MPαm\(P\)\(nC\)ϕm\(P\)\(Tr,xN2\)\\Delta P=\\sum\_\{m=1\}^\{M\_\{P\}\}\\alpha\_\{m\}^\{\(P\)\}\(n\_\{C\}\)\\phi\_\{m\}^\{\(P\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\) Represent Δy=∑m=1Myαm\(y\)\(nC\)ϕm\(y\)\(Tr,xN2\)\\Delta y=\\sum\_\{m=1\}^\{M\_\{y\}\}\\alpha\_\{m\}^\{\(y\)\}\(n\_\{C\}\)\\phi\_\{m\}^\{\(y\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\) Optimize the basis\-internal constants and the coefficient functions αm\(P\)\(nC\)\\alpha\_\{m\}^\{\(P\)\}\(n\_\{C\}\)for ΔP\\Delta Pusing gradient descent Optimize the basis\-internal constants and the coefficient functions αm\(y\)\(nC\)\\alpha\_\{m\}^\{\(y\)\}\(n\_\{C\}\)for Δy\\Delta yusing gradient descent ⊳\\trianglerightPR\-EOS correction Construct Pcorr=PPR\+ΔP\(Tr,xN2,nC\)P\_\{\\mathrm\{corr\}\}=P\_\{\\mathrm\{PR\}\}\+\\Delta P\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\) Construct ycorr=yPR\+Δy\(Tr,xN2,nC\)y\_\{\\mathrm\{corr\}\}=y\_\{\\mathrm\{PR\}\}\+\\Delta y\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\) return*Pcorr\(Tr,xN2,nC\)P\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)andycorr\(Tr,xN2,nC\)y\_\{\\mathrm\{corr\}\}\(T\_\{r\},x\_\{N\_\{2\}\},n\_\{C\}\)* Algorithm 2Regression using shared symbolic basis functionsTable 2:System\-specific symbolic expressions discovered for the pressure correction termΔP\\Delta Pin six n\-alkane \+ N2binary systems\. Highlighted subexpressions denote recurring symbolic forms identified across multiple systems\. The representative basis column indicates the symbolic basis functions selected for the subsequent multilevel symbolic regression\.SystemRelativeerror \[%\]RepresentativebasisDiscovered expression forΔP\(Tr,xN2\)\\Delta P\(T\_\{r\},x\_\{N\_\{2\}\}\)C5H120\.0200\.020ϕ1\(P\)\\phi\_\{1\}^\{\(P\)\}ΔP=a1\(xN2\+a2−ea3Tr\+xN2\)⏟\(A\) linear–exponential form\\displaystyle\\Delta P=a\_\{1\}\\underbrace\{\\left\(x\_\{N\_\{2\}\}\+a\_\{2\}\-e^\{a\_\{3\}T\_\{r\}\+x\_\{N\_\{2\}\}\}\\right\)\}\_\{\\text\{\(A\) linear\-\-exponential form\}\}C6H140\.0390\.039–ΔP=a1\(xN2\+a2−ea3xN2\)⏟\(A\) linear–exponential form\\displaystyle\\Delta P=a\_\{1\}\\underbrace\{\\left\(x\_\{N\_\{2\}\}\+a\_\{2\}\-e^\{a\_\{3\}x\_\{N\_\{2\}\}\}\\right\)\}\_\{\\text\{\(A\) linear\-\-exponential form\}\}C7H160\.0570\.057ϕ2\(P\)\\phi\_\{2\}^\{\(P\)\}ΔP=a1\(Tr\+xN22\)\(xN2\+a2\)⏟\(B\) coupled polynomial form\+a3\\displaystyle\\Delta P=a\_\{1\}\\underbrace\{\\left\(T\_\{r\}\+x\_\{N\_\{2\}\}^\{2\}\\right\)\\left\(x\_\{N\_\{2\}\}\+a\_\{2\}\\right\)\}\_\{\\text\{\(B\) coupled polynomial form\}\}\+a\_\{3\}C9H200\.0130\.013ϕ3\(P\)\\phi\_\{3\}^\{\(P\)\}ΔP=a1\(Tr\+xN2\+a2\)\(xN2\+a3\)⏟\(B\) coupled polynomial form\\displaystyle\\Delta P=a\_\{1\}\\underbrace\{\\left\(T\_\{r\}\+x\_\{N\_\{2\}\}\+a\_\{2\}\\right\)\\left\(x\_\{N\_\{2\}\}\+a\_\{3\}\\right\)\}\_\{\\text\{\(B\) coupled polynomial form\}\}C10H220\.0080\.008ϕ4\(P\)\\phi\_\{4\}^\{\(P\)\}ΔP=\(a1Tr\+a2xN2\)⏟\(B\) coupled polynomial form\(exN2\+a3\(Tr\+a4xN22\)\+a5\)⏟\(C\) exponential nonlinear form\+a6\\displaystyle\\Delta P=\\underbrace\{\\left\(a\_\{1\}T\_\{r\}\+a\_\{2\}x\_\{N\_\{2\}\}\\right\)\}\_\{\\text\{\(B\) coupled polynomial form\}\}\\,\\underbrace\{\\left\(e^\{x\_\{N\_\{2\}\}\+a\_\{3\}\(T\_\{r\}\+a\_\{4\}x\_\{N\_\{2\}\}^\{2\}\)\}\+a\_\{5\}\\right\)\}\_\{\\text\{\(C\) exponential nonlinear form\}\}\+a\_\{6\}C12H260\.1110\.111ϕ5\(P\)\\phi\_\{5\}^\{\(P\)\}ΔP=a2−Tr⏟\(D\) inverse\-temperature form\(xN2−exN2\+a1\)⏟\(A\) linear–exponential form\\displaystyle\\Delta P=\\underbrace\{a\_\{2\}^\{\-T\_\{r\}\}\}\_\{\\text\{\(D\) inverse\-temperature form\}\}\\,\\underbrace\{\\left\(x\_\{N\_\{2\}\}\-e^\{x\_\{N\_\{2\}\}\}\+a\_\{1\}\\right\)\}\_\{\\text\{\(A\) linear\-\-exponential form\}\}Table 3:System\-specific symbolic expressions discovered for the vapor\-composition correction termΔy\\Delta yin six n\-alkane \+ N2binary systems\. Highlighted subexpressions denote recurring symbolic forms identified across multiple systems\. The representative basis column indicates the symbolic basis functions selected for the subsequent multilevel symbolic regression\.SystemRelativeerror \[%\]RepresentativebasisDiscovered expression forΔy\(Tr,xN2\)\\Delta y\(T\_\{r\},x\_\{N\_\{2\}\}\)C5H120\.0910\.091ϕ1\(y\)\\phi\_\{1\}^\{\(y\)\}Δy=−xN2\(a1\+eTr\(a2xN2\+a3\)\)⏟\(A\) exponential nonlinear form\\displaystyle\\Delta y=\-x\_\{N\_\{2\}\}\\underbrace\{\\left\(a\_\{1\}\+e^\{T\_\{r\}\}\(a\_\{2\}x\_\{N\_\{2\}\}\+a\_\{3\}\)\\right\)\}\_\{\\text\{\(A\) exponential nonlinear form\}\}C6H140\.0660\.066ϕ2\(y\)\\phi\_\{2\}^\{\(y\)\}Δy=\(a1−xN2−ea2TrxN2\)⏟\(B\) linear–exponential form\\displaystyle\\Delta y=\\underbrace\{\\left\(a\_\{1\}\-x\_\{N\_\{2\}\}\-e^\{a\_\{2\}T\_\{r\}x\_\{N\_\{2\}\}\}\\right\)\}\_\{\\text\{\(B\) linear\-\-exponential form\}\}C7H160\.0720\.072ϕ3\(y\)\\phi\_\{3\}^\{\(y\)\}Δy=\(a1Tr\+a2\)\(xN2\+a3Tr2\)⏟\(C\) coupled polynomial form\\displaystyle\\Delta y=\\underbrace\{\\left\(a\_\{1\}T\_\{r\}\+a\_\{2\}\\right\)\\left\(x\_\{N\_\{2\}\}\+a\_\{3\}T\_\{r\}^\{2\}\\right\)\}\_\{\\text\{\(C\) coupled polynomial form\}\}C9H200\.0290\.029ϕ4\(y\)\\phi\_\{4\}^\{\(y\)\}Δy=\(Tr\+a1\)\(xN2Tr−a2\+a3\)⏟\(C\) coupled polynomial form\\displaystyle\\Delta y=\\underbrace\{\\left\(T\_\{r\}\+a\_\{1\}\\right\)\\left\(\\dfrac\{x\_\{N\_\{2\}\}\}\{T\_\{r\}\-a\_\{2\}\}\+a\_\{3\}\\right\)\}\_\{\\text\{\(C\) coupled polynomial form\}\}C10H220\.0140\.014ϕ5\(y\)\\phi\_\{5\}^\{\(y\)\}Δy=\(a1\+a2\(xN2−exN2\)⏟\(B\) linear–exponential formea3Tr\)\(a4\+a5xN2eTr\)⏟\(A\) exponential nonlinear form\+a6\\displaystyle\\Delta y=\\left\(a\_\{1\}\+a\_\{2\}\\hskip\-18\.0pt\\underbrace\{\\left\(x\_\{N\_\{2\}\}\-e^\{x\_\{N\_\{2\}\}\}\\right\)\}\_\{\\text\{\(B\) linear\-\-exponential form\}\}\\hskip\-18\.0pte^\{a\_\{3\}T\_\{r\}\}\\right\)\\underbrace\{\\left\(a\_\{4\}\+a\_\{5\}x\_\{N\_\{2\}\}e^\{T\_\{r\}\}\\right\)\}\_\{\\text\{\(A\) exponential nonlinear form\}\}\+a\_\{6\}C12H260\.0730\.073ϕ6\(y\)\\phi\_\{6\}^\{\(y\)\}Δy=a1xN2\(eTr\)a2xN2⏟\(A\) exponential nonlinear form\\displaystyle\\Delta y=a\_\{1\}x\_\{N\_\{2\}\}\\underbrace\{\\left\(e^\{T\_\{r\}\}\\right\)^\{a\_\{2\}^\{x\_\{N\_\{2\}\}\}\}\}\_\{\\text\{\(A\) exponential nonlinear form\}\}As a result of the system\-specific symbolic regression, distinct symbolic expressions are obtained for the pressure and vapor\-composition correction terms of each hydrocarbon–nitrogen system, as summarized in Tables[2](https://arxiv.org/html/2608.11255#S3.T2)and[3](https://arxiv.org/html/2608.11255#S3.T3)\. Although the discovered expressions differ in both analytical form and complexity, several recurring symbolic forms consistently emerge across multiple hydrocarbon systems, including linear–exponential, coupled polynomial, exponential, and inverse\-temperature forms\. This observation suggests that the underlying correction mechanisms are not entirely system specific but instead share common symbolic structures with different numerical parameterizations\. Motivated by this observation, the proposed multilevel symbolic regression extracts these recurring symbolic forms as shared basis functions while parameterizing only their coefficients as functions of the carbon number, thereby preserving analytical interpretability while enabling a unified symbolic correction model across multiple hydrocarbon–nitrogen systems\. \(a\)Polynomial interpolation of the coefficient functionsαm\\alpha\_\{m\}forΔP\\Delta P\. \(b\)Polynomial interpolation of the coefficient functionsαm\\alpha\_\{m\}forΔy\\Delta y\. Figure 2:Polynomial interpolation of the learned systemwise coefficient values used in the shared\-basis multilevel model\. Symbols denote the coefficient values extracted from the inverse model, and solid curves denote the corresponding polynomial interpolants as functions of carbon numbernCn\_\{C\}\.To enable prediction for different hydrocarbon systems using a common symbolic basis, the system\-specific coefficients are parameterized as continuous functions of the carbon number\. The coefficient values identified for each binary system are first extracted from the multilevel symbolic regression model and subsequently approximated using polynomial interpolation, as illustrated in Fig\.[2](https://arxiv.org/html/2608.11255#S3.F2)\. Separate interpolants are constructed for the pressure correction coefficients, as shown in Fig\.[2](https://arxiv.org/html/2608.11255#S3.F2)a, and for the vapor\-composition correction coefficients, as shown in Fig\.[2](https://arxiv.org/html/2608.11255#S3.F2)b\. As a result, only the coefficient values vary systematically with the carbon number, whereas the shared symbolic basis functions remain unchanged\. This formulation enables the proposed model to represent multiple hydrocarbon–nitrogen systems within a unified symbolic expression while preserving the interpretability of the discovered symbolic correction model\. Figure 3:Pressure\-composition diagrams for the C5H12\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.Figure 4:Pressure\-composition diagrams for the C6H14\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.Figure 5:Pressure\-composition diagrams for the C7H16\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.Figure 6:Pressure\-composition diagrams for the C9H20\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.Figure 7:Pressure\-composition diagrams for the C10H22\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.Figure 8:Pressure\-composition diagrams for the C12H26\+ N2system at different reduced temperatures\. The experimental VLE curves, the PR\-EOS predictions, and the multilevel regression predictions are compared in each temperature panel\.To evaluate the predictive performance of the proposed multilevel symbolic regression model, the corrected PR\-EOS predictions are compared with both the experimental VLE measurements and the original PR\-EOS for all six hydrocarbon–nitrogen systems over a wide range of reduced temperatures, as shown in Figs\.[3](https://arxiv.org/html/2608.11255#S3.F3)–[8](https://arxiv.org/html/2608.11255#S3.F8)\. Overall, the proposed symbolic correction consistently improves agreement with the experimental pressure–composition curves while preserving the physically consistent trends predicted by the underlying PR\-EOS\. Across all hydrocarbon systems, the corrected model accurately reproduces the pressure variation over the entire composition range and substantially reduces the systematic deviations observed in the original PR\-EOS, demonstrating that the learned symbolic corrections effectively capture the dominant modeling errors\. For the lighter hydrocarbon systems, including C5H12, C6H14, and C7H16, the proposed model accurately reproduces the experimental phase\-equilibrium curves over all investigated reduced temperatures, as shown in Figs\.[3](https://arxiv.org/html/2608.11255#S3.F3)–[5](https://arxiv.org/html/2608.11255#S3.F5)\. Although the original PR\-EOS already provides reasonable agreement with the experimental data, noticeable discrepancies remain, particularly in regions exhibiting strong nonlinear pressure variation\. The proposed symbolic correction consistently reduces these deviations while preserving the smooth thermodynamic trends of the original equation of state, resulting in excellent agreement with the measured VLE curves throughout the entire composition range\. The improvement becomes even more evident for the heavier hydrocarbon systems, including C9H20, C10H22, and C12H26, as illustrated in Figs\.[6](https://arxiv.org/html/2608.11255#S3.F6)–[8](https://arxiv.org/html/2608.11255#S3.F8)\. For these systems, the original PR\-EOS exhibits larger systematic deviations from the experimental measurements, particularly at elevated pressures\. By incorporating the learned symbolic correction, the proposed model substantially improves the agreement with the experimental pressure–composition curves while maintaining smooth and physically consistent predictions across different reduced temperatures\. These results demonstrate that the proposed multilevel symbolic regression successfully captures systematic PR\-EOS errors over a broad range of hydrocarbon chain lengths using a unified symbolic correction model\. Table 4:Accuracy comparison for pressure prediction for the hydrocarbon–nitrogen VLE dataset\. Train and test denote the common pointwise 80/20 split used for evaluation\. Reported metrics are the mean squared error \(MSE\), coefficient of determination \(R2R^\{2\}\), maximum absolute error, and mean absolute error\. The best value in each column is shown in bold\.ModelSplitMSER2R^\{2\}max\|P−P^\|\\max\|P\-\\hat\{P\}\|mean\|P−P^\|\\mathrm\{mean\}\\,\|P\-\\hat\{P\}\|Two\-level SR \(proposed\)Train1\.1085×𝟏𝟎−𝟏\\mathbf\{1\.1085\\times 10^\{\-1\}\}0\.9995\\mathbf\{0\.9995\}2\.0615×𝟏𝟎𝟎\\mathbf\{2\.0615\\times 10^\{0\}\}2\.5215×𝟏𝟎−𝟏\\mathbf\{2\.5215\\times 10^\{\-1\}\}Test1\.2279×𝟏𝟎−𝟏\\mathbf\{1\.2279\\times 10^\{\-1\}\}0\.9994\\mathbf\{0\.9994\}8\.1785×𝟏𝟎−𝟏\\mathbf\{8\.1785\\times 10^\{\-1\}\}2\.8475×𝟏𝟎−𝟏\\mathbf\{2\.8475\\times 10^\{\-1\}\}SRTrain1\.4301×1001\.4301\\times 10^\{0\}0\.99410\.99415\.9095×1005\.9095\\times 10^\{0\}7\.5660×10−17\.5660\\times 10^\{\-1\}Test7\.3063×10−17\.3063\\times 10^\{\-1\}0\.99630\.99632\.8557×1002\.8557\\times 10^\{0\}6\.1370×10−16\.1370\\times 10^\{\-1\}PR\-EOSTrain1\.2277×1011\.2277\\times 10^\{1\}0\.94890\.94892\.1876×1012\.1876\\times 10^\{1\}2\.0157×1002\.0157\\times 10^\{0\}Test9\.3985×1009\.3985\\times 10^\{0\}0\.95280\.95281\.2690×1011\.2690\\times 10^\{1\}1\.8271×1001\.8271\\times 10^\{0\}To quantitatively evaluate the effectiveness of the proposed multilevel symbolic regression, Table[4](https://arxiv.org/html/2608.11255#S3.T4)compares its pressure prediction accuracy with those of the conventional symbolic regression model and the original PR\-EOS using a common 80/20 train–test split\. The proposed two\-level symbolic regression consistently achieves the lowest prediction errors and the highest coefficient of determination on both the training and test sets\. On the test set, the mean squared error is reduced from7\.31×10−17\.31\\times 10^\{\-1\}for the conventional symbolic regression to1\.23×10−11\.23\\times 10^\{\-1\}, while the mean absolute error decreases from6\.14×10−16\.14\\times 10^\{\-1\}to2\.85×10−12\.85\\times 10^\{\-1\}\. Compared with the original PR\-EOS, the improvement is even more substantial, reducing the mean squared error by nearly two orders of magnitude\. These results demonstrate that introducing shared symbolic basis functions together with carbon\-number\-dependent coefficient functions significantly improves prediction accuracy without sacrificing generalization performance\. Table 5:Accuracy comparison for vapor\-phase nitrogen composition prediction for the hydrocarbon–nitrogen VLE dataset\. Train and test denote the common pointwise 80/20 split used for evaluation\. Reported metrics are the mean squared error \(MSE\), coefficient of determination \(R2R^\{2\}\), maximum absolute error, and mean absolute error\. The best value in each column is shown in bold\.ModelSplitMSER2R^\{2\}max\|yN2−y^N2\|\\max\|y\_\{N\_\{2\}\}\-\\hat\{y\}\_\{N\_\{2\}\}\|mean\|yN2−y^N2\|\\mathrm\{mean\}\\,\|y\_\{N\_\{2\}\}\-\\hat\{y\}\_\{N\_\{2\}\}\|Two\-level SR \(proposed\)Train5\.6785×𝟏𝟎−𝟓\\mathbf\{5\.6785\\times 10^\{\-5\}\}0\.9987\\mathbf\{0\.9987\}5\.7109×𝟏𝟎−𝟐\\mathbf\{5\.7109\\times 10^\{\-2\}\}4\.7903×𝟏𝟎−𝟑\\mathbf\{4\.7903\\times 10^\{\-3\}\}Test3\.7679×𝟏𝟎−𝟓\\mathbf\{3\.7679\\times 10^\{\-5\}\}0\.9986\\mathbf\{0\.9986\}1\.1891×𝟏𝟎−𝟐\\mathbf\{1\.1891\\times 10^\{\-2\}\}5\.0269×𝟏𝟎−𝟑\\mathbf\{5\.0269\\times 10^\{\-3\}\}SRTrain1\.2469×10−41\.2469\\times 10^\{\-4\}0\.99710\.99716\.6379×10−26\.6379\\times 10^\{\-2\}7\.4529×10−37\.4529\\times 10^\{\-3\}Test4\.8733×10−54\.8733\\times 10^\{\-5\}0\.99820\.99821\.5917×10−21\.5917\\times 10^\{\-2\}5\.0378×10−35\.0378\\times 10^\{\-3\}PR\-EOSTrain1\.0735×10−31\.0735\\times 10^\{\-3\}0\.97500\.97501\.1880×10−11\.1880\\times 10^\{\-1\}2\.1437×10−22\.1437\\times 10^\{\-2\}Test6\.4073×10−46\.4073\\times 10^\{\-4\}0\.97620\.97625\.6650×10−25\.6650\\times 10^\{\-2\}1\.7970×10−21\.7970\\times 10^\{\-2\}To verify that the proposed multilevel symbolic regression accurately predicts the equilibrium composition in addition to pressure, Table[5](https://arxiv.org/html/2608.11255#S3.T5)summarizes the prediction accuracy for the vapor\-phase nitrogen mole fraction\. Similar to the pressure prediction results, the proposed model consistently achieves the lowest MSE and the highestR2R^\{2\}on both the training and test sets\. Compared with the original PR\-EOS, the prediction errors are substantially reduced, while modest but consistent improvements over the conventional symbolic regression further demonstrate the benefit of the proposed two\-level symbolic formulation\. \(a\)C5H12\+ N2 \(b\)C6H14\+ N2 \(c\)C7H16\+ N2 \(d\)C9H20\+ N2 \(e\)C10H22\+ N2 \(f\)C12H26\+ N2 Figure 9:Actual\-versus\-predicted pressure plots for the six hydrocarbon–nitrogen systems\. Black circles denote the original PR\-EOS predictions, red circles denote corrected training samples, green squares denote corrected test samples, and the black line represents the identity line\.To further evaluate the pressure prediction accuracy, the actual and predicted pressures are compared for all six hydrocarbon–nitrogen systems in Fig\.[9](https://arxiv.org/html/2608.11255#S3.F9)\. Compared with the original PR\-EOS predictions, the proposed symbolic correction substantially reduces the prediction error, with both the training and test samples becoming much more closely aligned with the identity line\. The improvement is consistently observed across all six hydrocarbon systems, indicating that the proposed correction effectively captures the systematic pressure deviations of the PR\-EOS while maintaining strong predictive performance on unseen test samples\. \(a\)C5H12\+ N2 \(b\)C6H14\+ N2 \(c\)C7H16\+ N2 \(d\)C9H20\+ N2 \(e\)C10H22\+ N2 \(f\)C12H26\+ N2 Figure 10:Vapor composition parity plots for the lookup coefficient model across the six hydrocarbon–nitrogen systems\. black circles denote the original PR\-EOS predictions, red circles denote corrected training samples, magenta squares denote corrected test samples, and the black line denotes the identity line\.Figure[10](https://arxiv.org/html/2608.11255#S3.F10)presents the vapor composition parity plots for all six hydrocarbon–nitrogen systems\. Compared with the original PR\-EOS predictions, the proposed lookup coefficient model significantly improves the prediction accuracy, with both the training and test samples closely distributed along the identity line\. The consistent agreement across all systems demonstrates that the proposed correction model accurately captures the systematic vapor composition deviations of the PR\-EOS while maintaining good generalization performance\. Figure 11:Actual\-versus\-predicted plots for pressure \(left\) and vapor mole fraction \(right\) obtained by combining all six hydrocarbon–nitrogen systems\. Black circles denote the original PR\-EOS predictions, red crosses denote corrected training samples, green squares denote corrected test samples, and the solid black line represents the identity line\.To provide an overall assessment of the proposed symbolic correction model, Fig\.[11](https://arxiv.org/html/2608.11255#S3.F11)combines the prediction results from all six hydrocarbon–nitrogen systems into unified actual\-versus\-predicted plots for pressure and vapor mole fraction\. Compared with the original PR\-EOS predictions, the corrected training and test samples exhibit substantially improved agreement with the identity line for both quantities\. In particular, the pressure predictions are significantly improved over the entire pressure range, while the vapor\-phase composition predictions remain highly accurate across a broad range of mixture compositions\. The close agreement observed for both the training and test samples demonstrates that the proposed multilevel symbolic regression consistently improves predictive accuracy while maintaining robust generalization across different hydrocarbon systems\.    Figure 12:Systemwise comparison of prediction errors for pressure and vapor composition across the six hydrocarbon–nitrogen mixtures\. The top panel shows the absolute relative error, the middle panel shows the root\-mean\-square error \(RMSE\), and the bottom panel shows the mean absolute error \(MAE\)\. In each case, the original PR\-EOS predictions, the pooled symbolic regression model, and the proposed multilevel regression model are compared for every system\.To assess the effectiveness of the proposed multilevel symbolic regression, Fig\.[12](https://arxiv.org/html/2608.11255#S3.F12)compares the prediction errors of the original PR\-EOS, the conventional symbolic regression model, and the proposed two\-level symbolic regression model across all six hydrocarbon–nitrogen systems\. The absolute relative error, RMSE, and MAE are reported for both pressure and vapor composition\. Compared with the original PR\-EOS, both symbolic regression models substantially reduce the prediction errors\. More importantly, the proposed two\-level symbolic regression consistently outperforms the conventional symbolic regression across nearly all hydrocarbon systems and error metrics\. These results demonstrate that separating system\-specific symbolic discovery from carbon\-number\-dependent coefficient learning provides a more accurate symbolic representation than learning a single symbolic expression from the pooled dataset\. Figure 13:Representative pressure–composition phase diagram for the C7H16/N2system\. The experimentally estimated critical points and the corresponding reference critical locus are included for visualization\. The left panel shows the original PR\-EOS prediction, whereas the right panel shows the symbolically corrected PR\-EOS prediction\.As a representative example, Fig\.[13](https://arxiv.org/html/2608.11255#S3.F13)presents the pressure–composition phase diagram for the C7H16/N2system together with the experimentally estimated critical points and a reference critical locus\. The reference critical locus is included solely to facilitate visualization of the overall phase behavior\. Compared with the original PR\-EOS, the symbolically corrected PR\-EOS more closely follows the experimental phase envelope while preserving the overall thermodynamic trends\. Improved agreement is also observed in the vicinity of the estimated critical region, further demonstrating that the proposed symbolic correction enhances the predictive capability of the PR\-EOS without altering its underlying thermodynamic structure\. ## 4Conclusion In this work, we proposed a multilevel symbolic regression framework for correcting the Peng–Robinson equation of state for hydrocarbon–nitrogen vapor–liquid equilibrium prediction\. System\-specific symbolic correction terms were first identified from experimental data, and recurring symbolic structures were subsequently organized into a shared symbolic basis with coefficient functions parameterized by the carbon number\. The resulting model provides compact and interpretable symbolic correction expressions while improving prediction accuracy for both pressure and vapor composition across multiple hydrocarbon–nitrogen systems\. Numerical results demonstrated that the proposed two\-level symbolic regression consistently outperformed both the original PR\-EOS and a conventional symbolic regression model in terms of prediction accuracy\. The proposed correction also produced improved agreement with the experimental pressure–composition diagrams and representative phase envelopes while preserving the overall thermodynamic behavior of the underlying PR\-EOS\. The present study considered symbolic corrections to the PR\-EOS for hydrocarbon–nitrogen mixtures\. Future work will investigate equation\-level symbolic corrections that directly modify the analytical form of the equation of state while preserving physical consistency\. ## References - \[1\]R\. Abedini, I\. Zanganeh, and M\. 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Table 6:Learned internal constants for the shared pressure basis functions\.BasisParametersValuesϕ1\(P\)\\phi\_\{1\}^\{\(P\)\}\(a1\(P\),a2\(P\),a3\(P\)\)\(a\_\{1\}^\{\(P\)\},a\_\{2\}^\{\(P\)\},a\_\{3\}^\{\(P\)\}\)\(0\.714,1\.60,−1\.30\)\(0\.714,\\ 1\.60,\\ \-1\.30\)ϕ2\(P\)\\phi\_\{2\}^\{\(P\)\}\(b1\(P\),b2\(P\),b3\(P\)\)\(b\_\{1\}^\{\(P\)\},b\_\{2\}^\{\(P\)\},b\_\{3\}^\{\(P\)\}\)\(−0\.397,2\.01,0\.449\)\(\-0\.397,\\ 2\.01,\\ 0\.449\)ϕ3\(P\)\\phi\_\{3\}^\{\(P\)\}\(c1\(P\),c2\(P\),c3\(P\)\)\(c\_\{1\}^\{\(P\)\},c\_\{2\}^\{\(P\)\},c\_\{3\}^\{\(P\)\}\)\(−0\.656,2\.13,0\.649\)\(\-0\.656,\\ 2\.13,\\ 0\.649\)ϕ4\(P\)\\phi\_\{4\}^\{\(P\)\}\(d1\(P\),d2\(P\),d3\(P\),d4\(P\),d5\(P\),d6\(P\)\)\(d\_\{1\}^\{\(P\)\},d\_\{2\}^\{\(P\)\},d\_\{3\}^\{\(P\)\},d\_\{4\}^\{\(P\)\},d\_\{5\}^\{\(P\)\},d\_\{6\}^\{\(P\)\}\)\(0\.739,−0\.388,−0\.173,0\.052,−0\.086,0\.316\)\(0\.739,\\ \-0\.388,\\ \-0\.173,\\ 0\.052,\\ \-0\.086,\\ 0\.316\)ϕ5\(P\)\\phi\_\{5\}^\{\(P\)\}\(e1\(P\),e2\(P\)\)\(e\_\{1\}^\{\(P\)\},e\_\{2\}^\{\(P\)\}\)\(1\.88,2\.31\)\(1\.88,\\ 2\.31\)Table 7:Learned internal constants for the shared vapor\-composition basis functions\.BasisParametersValuesϕ1\(y\)\\phi\_\{1\}^\{\(y\)\}\(a1\(y\),a2\(y\),a3\(y\)\)\(a\_\{1\}^\{\(y\)\},a\_\{2\}^\{\(y\)\},a\_\{3\}^\{\(y\)\}\)\(0\.610,0\.219,0\.696\)\(0\.610,\\ 0\.219,\\ 0\.696\)ϕ2\(y\)\\phi\_\{2\}^\{\(y\)\}\(b1\(y\),b2\(y\)\)\(b\_\{1\}^\{\(y\)\},b\_\{2\}^\{\(y\)\}\)\(1\.09,0\.443\)\(1\.09,\\ 0\.443\)ϕ3\(y\)\\phi\_\{3\}^\{\(y\)\}\(c1\(y\),c2\(y\),c3\(y\)\)\(c\_\{1\}^\{\(y\)\},c\_\{2\}^\{\(y\)\},c\_\{3\}^\{\(y\)\}\)\(0\.672,0\.914,0\.254\)\(0\.672,\\ 0\.914,\\ 0\.254\)ϕ4\(y\)\\phi\_\{4\}^\{\(y\)\}\(d1\(y\),d2\(y\),d3\(y\)\)\(d\_\{1\}^\{\(y\)\},d\_\{2\}^\{\(y\)\},d\_\{3\}^\{\(y\)\}\)\(1\.34,2\.42,0\.127\)\(1\.34,\\ 2\.42,\\ 0\.127\)ϕ5\(y\)\\phi\_\{5\}^\{\(y\)\}\(e1\(y\),e2\(y\),e3\(y\),e4\(y\),e5\(y\),e6\(y\)\)\(e\_\{1\}^\{\(y\)\},e\_\{2\}^\{\(y\)\},e\_\{3\}^\{\(y\)\},e\_\{4\}^\{\(y\)\},e\_\{5\}^\{\(y\)\},e\_\{6\}^\{\(y\)\}\)\(0\.605,0\.279,1\.10,0\.561,−0\.217,−0\.028\)\(0\.605,\\ 0\.279,\\ 1\.10,\\ 0\.561,\\ \-0\.217,\\ \-0\.028\)ϕ6\(y\)\\phi\_\{6\}^\{\(y\)\}\(g1\(y\),g2\(y\)\)\(g\_\{1\}^\{\(y\)\},g\_\{2\}^\{\(y\)\}\)\(−0\.353,1\.16\)\(\-0\.353,\\ 1\.16\)Tables[8](https://arxiv.org/html/2608.11255#A1.T8)and[9](https://arxiv.org/html/2608.11255#A1.T9)report the system\-specific coefficients associated with each shared basis function\. These coefficients determine the contribution of each basis function to the correction model for individual hydrocarbon–nitrogen systems and constitute the reference data used for the coefficient interpolation described in Section 3\. For completeness and reproducibility, all optimized basis constants and system\-specific coefficients are provided explicitly\. Together with the symbolic basis functions presented in the main text, these parameters fully specify the proposed multilevel symbolic regression model and enable direct reconstruction of the pressure and vapor\-composition correction functions\. Table 8:System\-specific coefficientsαm\(P,s\)\\alpha\_\{m\}^\{\(P,s\)\}for the multilevel pressure modelΔP\(s\)\(Tr,xN2\)=∑m=15αm\(P,s\)ϕm\(P\)\(Tr,xN2\)\\Delta P^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)=\\sum\_\{m=1\}^\{5\}\\alpha\_\{m\}^\{\(P,s\)\}\\phi\_\{m\}^\{\(P\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)\.Systemα1\(P,s\)\\alpha\_\{1\}^\{\(P,s\)\}α2\(P,s\)\\alpha\_\{2\}^\{\(P,s\)\}α3\(P,s\)\\alpha\_\{3\}^\{\(P,s\)\}α4\(P,s\)\\alpha\_\{4\}^\{\(P,s\)\}α5\(P,s\)\\alpha\_\{5\}^\{\(P,s\)\}C5/N2−0\.039\-0\.0390\.3580\.3580\.1440\.1440\.2160\.216−0\.052\-0\.052C6/N2−0\.095\-0\.0950\.3480\.3480\.1480\.1480\.2600\.2600\.0030\.003C7/N2−0\.176\-0\.1760\.3560\.3560\.1580\.1580\.3310\.3310\.0140\.014C9/N2−0\.342\-0\.3420\.1750\.1750\.3870\.3870\.4370\.437−0\.084\-0\.084C10/N2−0\.328\-0\.3280\.2450\.2450\.1070\.1070\.6570\.6570\.0550\.055C12/N2−0\.435\-0\.4350\.3280\.3280\.2210\.2210\.6760\.676−0\.165\-0\.165Table 9:System\-specific coefficientsαm\(y,s\)\\alpha\_\{m\}^\{\(y,s\)\}for the multilevel vapor\-composition modelΔy\(s\)\(Tr,xN2\)=∑m=16αm\(y,s\)ϕm\(y\)\(Tr,xN2\)\\Delta y^\{\(s\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)=\\sum\_\{m=1\}^\{6\}\\alpha\_\{m\}^\{\(y,s\)\}\\phi\_\{m\}^\{\(y\)\}\(T\_\{r\},x\_\{N\_\{2\}\}\)\.Systemα1\(y,s\)\\alpha\_\{1\}^\{\(y,s\)\}α2\(y,s\)\\alpha\_\{2\}^\{\(y,s\)\}α3\(y,s\)\\alpha\_\{3\}^\{\(y,s\)\}α4\(y,s\)\\alpha\_\{4\}^\{\(y,s\)\}α5\(y,s\)\\alpha\_\{5\}^\{\(y,s\)\}α6\(y,s\)\\alpha\_\{6\}^\{\(y,s\)\}C5/N20\.1220\.1220\.1990\.1990\.1510\.1511\.111\.11−1\.03\-1\.030\.3790\.379C6/N20\.0940\.0940\.1920\.1920\.0770\.0771\.151\.15−1\.01\-1\.010\.3870\.387C7/N20\.0770\.0770\.1780\.1780\.0030\.0031\.131\.13−0\.895\-0\.8950\.3900\.390C9/N20\.0740\.074−0\.032\-0\.032−0\.008\-0\.0080\.6220\.622−0\.038\-0\.0380\.3150\.315C10/N20\.0220\.022−0\.052\-0\.052−0\.013\-0\.0130\.2590\.2590\.1910\.1910\.2560\.256C12/N20\.2330\.2330\.1410\.141−0\.124\-0\.124−0\.189\-0\.1890\.4480\.4480\.1950\.195 ## Appendix BExplicit forms of the multilevel correction model This appendix summarizes the explicit form of the multilevel correction model used in the main text\. The model is constructed from shared symbolic basis functions together with system\-specific coefficients\. For each hydrocarbon–nitrogen system, the pressure correction and the vapor\-composition correction are represented as linear combinations of the corresponding shared basis functions\. The corrected predictions are then obtained by adding these correction terms to the original PR\-EOS predictions\. For notational simplicity, we write T≡Tr,x≡xN2\.T\\equiv T\_\{r\},\\qquad x\\equiv x\_\{N\_\{2\}\}\.With this notation, the corrected pressure and vapor composition for systemssare given by Pcorr\(s\)\(T,x\)=PPR\+ΔP\(s\)\(T,x\),ycorr\(s\)\(T,x\)=yPR\+Δy\(s\)\(T,x\)\.P\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T,x\)=P\_\{\\mathrm\{PR\}\}\+\\Delta P^\{\(s\)\}\(T,x\),\\qquad y\_\{\\mathrm\{corr\}\}^\{\(s\)\}\(T,x\)=y\_\{\\mathrm\{PR\}\}\+\\Delta y^\{\(s\)\}\(T,x\)\. The pressure correction is expressed as ΔP\(s\)\(T,x\)=∑m=15αm\(P,s\)ϕm\(P\)\(T,x\),\\Delta P^\{\(s\)\}\(T,x\)=\\sum\_\{m=1\}^\{5\}\\alpha\_\{m\}^\{\(P,s\)\}\\phi\_\{m\}^\{\(P\)\}\(T,x\),whereϕm\(P\)\\phi\_\{m\}^\{\(P\)\}denote the shared basis functions for pressure andαm\(P,s\)\\alpha\_\{m\}^\{\(P,s\)\}are the system\-specific coefficients\. Similarly, the vapor\-composition correction is expressed as Δy\(s\)\(T,x\)=∑m=16αm\(y,s\)ϕm\(y\)\(T,x\),\\Delta y^\{\(s\)\}\(T,x\)=\\sum\_\{m=1\}^\{6\}\\alpha\_\{m\}^\{\(y,s\)\}\\phi\_\{m\}^\{\(y\)\}\(T,x\),whereϕm\(y\)\\phi\_\{m\}^\{\(y\)\}denote the shared basis functions for vapor composition andαm\(y,s\)\\alpha\_\{m\}^\{\(y,s\)\}are the corresponding system\-specific coefficients\. ### B\.1Shared basis functions forΔP\\Delta P The shared pressure basis functions are ϕ1\(P\)\(T,x\)\\displaystyle\\phi\_\{1\}^\{\(P\)\}\(T,x\)=0\.714\(x\+1\.60−e−1\.30T\+x\),\\displaystyle=0\.714\\left\(x\+1\.60\-e^\{\-1\.30T\+x\}\\right\),\(16\)ϕ2\(P\)\(T,x\)\\displaystyle\\phi\_\{2\}^\{\(P\)\}\(T,x\)=−0\.397\(T\+x2\)\(x\+2\.01\)\+0\.449,\\displaystyle=\-0\.397\\,\(T\+x^\{2\}\)\(x\+2\.01\)\+0\.449,\(17\)ϕ3\(P\)\(T,x\)\\displaystyle\\phi\_\{3\}^\{\(P\)\}\(T,x\)=−0\.656\(T\+x−2\.13\)\(x\+0\.649\),\\displaystyle=\-0\.656\\,\(T\+x\-2\.13\)\(x\+0\.649\),\(18\)ϕ4\(P\)\(T,x\)\\displaystyle\\phi\_\{4\}^\{\(P\)\}\(T,x\)=\(0\.739T−0\.388x\)\(ex−0\.173\(T\+0\.052x2\)\+0\.086\)\+0\.316,\\displaystyle=\\left\(0\.739T\-0\.388x\\right\)\\left\(e^\{\\,x\-0\.173\(T\+0\.052x^\{2\}\)\}\+0\.086\\right\)\+0\.316,\(19\)ϕ5\(P\)\(T,x\)\\displaystyle\\phi\_\{5\}^\{\(P\)\}\(T,x\)=x−ex\+1\.88\(2\.31\)T\.\\displaystyle=\\frac\{x\-e^\{x\}\+1\.88\}\{\(2\.31\)^\{T\}\}\.\(20\) Using these basis functions, the system\-specific pressure corrections are written as ΔP\(C5\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C5\}\)\}\(T,x\)=−0\.039ϕ1\(P\)\+0\.358ϕ2\(P\)\+0\.144ϕ3\(P\)\+0\.216ϕ4\(P\)−0\.052ϕ5\(P\),\\displaystyle=\-0\.039\\,\\phi\_\{1\}^\{\(P\)\}\+0\.358\\,\\phi\_\{2\}^\{\(P\)\}\+0\.144\\,\\phi\_\{3\}^\{\(P\)\}\+0\.216\\,\\phi\_\{4\}^\{\(P\)\}\-0\.052\\,\\phi\_\{5\}^\{\(P\)\},\(21\)ΔP\(C6\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C6\}\)\}\(T,x\)=−0\.095ϕ1\(P\)\+0\.348ϕ2\(P\)\+0\.148ϕ3\(P\)\+0\.260ϕ4\(P\)\+0\.003ϕ5\(P\),\\displaystyle=\-0\.095\\,\\phi\_\{1\}^\{\(P\)\}\+0\.348\\,\\phi\_\{2\}^\{\(P\)\}\+0\.148\\,\\phi\_\{3\}^\{\(P\)\}\+0\.260\\,\\phi\_\{4\}^\{\(P\)\}\+0\.003\\,\\phi\_\{5\}^\{\(P\)\},\(22\)ΔP\(C7\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C7\}\)\}\(T,x\)=−0\.176ϕ1\(P\)\+0\.356ϕ2\(P\)\+0\.158ϕ3\(P\)\+0\.331ϕ4\(P\)\+0\.014ϕ5\(P\),\\displaystyle=\-0\.176\\,\\phi\_\{1\}^\{\(P\)\}\+0\.356\\,\\phi\_\{2\}^\{\(P\)\}\+0\.158\\,\\phi\_\{3\}^\{\(P\)\}\+0\.331\\,\\phi\_\{4\}^\{\(P\)\}\+0\.014\\,\\phi\_\{5\}^\{\(P\)\},\(23\)ΔP\(C9\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C9\}\)\}\(T,x\)=−0\.342ϕ1\(P\)\+0\.175ϕ2\(P\)\+0\.387ϕ3\(P\)\+0\.437ϕ4\(P\)−0\.084ϕ5\(P\),\\displaystyle=\-0\.342\\,\\phi\_\{1\}^\{\(P\)\}\+0\.175\\,\\phi\_\{2\}^\{\(P\)\}\+0\.387\\,\\phi\_\{3\}^\{\(P\)\}\+0\.437\\,\\phi\_\{4\}^\{\(P\)\}\-0\.084\\,\\phi\_\{5\}^\{\(P\)\},\(24\)ΔP\(C10\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C10\}\)\}\(T,x\)=−0\.328ϕ1\(P\)\+0\.245ϕ2\(P\)\+0\.107ϕ3\(P\)\+0\.657ϕ4\(P\)\+0\.055ϕ5\(P\),\\displaystyle=\-0\.328\\,\\phi\_\{1\}^\{\(P\)\}\+0\.245\\,\\phi\_\{2\}^\{\(P\)\}\+0\.107\\,\\phi\_\{3\}^\{\(P\)\}\+0\.657\\,\\phi\_\{4\}^\{\(P\)\}\+0\.055\\,\\phi\_\{5\}^\{\(P\)\},\(25\)ΔP\(C12\)\(T,x\)\\displaystyle\\Delta P^\{\(\\mathrm\{C12\}\)\}\(T,x\)=−0\.435ϕ1\(P\)\+0\.328ϕ2\(P\)\+0\.221ϕ3\(P\)\+0\.676ϕ4\(P\)−0\.165ϕ5\(P\)\.\\displaystyle=\-0\.435\\,\\phi\_\{1\}^\{\(P\)\}\+0\.328\\,\\phi\_\{2\}^\{\(P\)\}\+0\.221\\,\\phi\_\{3\}^\{\(P\)\}\+0\.676\\,\\phi\_\{4\}^\{\(P\)\}\-0\.165\\,\\phi\_\{5\}^\{\(P\)\}\.\(26\) ### B\.2Shared basis functions forΔy\\Delta y The shared basis functions for the vapor\-composition correction are ϕ1\(y\)\(T,x\)\\displaystyle\\phi\_\{1\}^\{\(y\)\}\(T,x\)=−x\(0\.610\+eT\(0\.219x\+0\.696\)\),\\displaystyle=\-x\\left\(0\.610\+e^\{T\}\(0\.219x\+0\.696\)\\right\),\(27\)ϕ2\(y\)\(T,x\)\\displaystyle\\phi\_\{2\}^\{\(y\)\}\(T,x\)=1\.09−x−e0\.443Tx,\\displaystyle=1\.09\-x\-e^\{0\.443Tx\},\(28\)ϕ3\(y\)\(T,x\)\\displaystyle\\phi\_\{3\}^\{\(y\)\}\(T,x\)=\(0\.672T−0\.914\)\(x\+0\.254T2\),\\displaystyle=\(0\.672T\-0\.914\)\(x\+0\.254T^\{2\}\),\(29\)ϕ4\(y\)\(T,x\)\\displaystyle\\phi\_\{4\}^\{\(y\)\}\(T,x\)=\(T\+1\.34\)\(xT−2\.42\+0\.127\),\\displaystyle=\(T\+1\.34\)\\left\(\\frac\{x\}\{T\-2\.42\}\+0\.127\\right\),\(30\)ϕ5\(y\)\(T,x\)\\displaystyle\\phi\_\{5\}^\{\(y\)\}\(T,x\)=\(0\.605−0\.279\(x−ex\)e1\.10T\)\(0\.561−0\.217xeT\)\+0\.028,\\displaystyle=\\left\(0\.605\-0\.279\(x\-e^\{x\}\)e^\{1\.10T\}\\right\)\\left\(0\.561\-0\.217xe^\{T\}\\right\)\+0\.028,\(31\)ϕ6\(y\)\(T,x\)\\displaystyle\\phi\_\{6\}^\{\(y\)\}\(T,x\)=−0\.353x\(eT\)\(1\.16\)x\.\\displaystyle=\-0\.353\\,x\\,\(e^\{T\}\)^\{\(1\.16\)^\{x\}\}\.\(32\) The corresponding system\-specific vapor\-composition corrections are Δy\(C5\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C5\}\)\}\(T,x\)=0\.122ϕ1\(y\)\+0\.199ϕ2\(y\)\+0\.151ϕ3\(y\)\+1\.11ϕ4\(y\)−1\.03ϕ5\(y\)\+0\.379ϕ6\(y\),\\displaystyle=0\.122\\,\\phi\_\{1\}^\{\(y\)\}\+0\.199\\,\\phi\_\{2\}^\{\(y\)\}\+0\.151\\,\\phi\_\{3\}^\{\(y\)\}\+1\.11\\,\\phi\_\{4\}^\{\(y\)\}\-1\.03\\,\\phi\_\{5\}^\{\(y\)\}\+0\.379\\,\\phi\_\{6\}^\{\(y\)\},\(33\)Δy\(C6\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C6\}\)\}\(T,x\)=0\.094ϕ1\(y\)\+0\.192ϕ2\(y\)\+0\.077ϕ3\(y\)\+1\.15ϕ4\(y\)−1\.01ϕ5\(y\)\+0\.387ϕ6\(y\),\\displaystyle=0\.094\\,\\phi\_\{1\}^\{\(y\)\}\+0\.192\\,\\phi\_\{2\}^\{\(y\)\}\+0\.077\\,\\phi\_\{3\}^\{\(y\)\}\+1\.15\\,\\phi\_\{4\}^\{\(y\)\}\-1\.01\\,\\phi\_\{5\}^\{\(y\)\}\+0\.387\\,\\phi\_\{6\}^\{\(y\)\},\(34\)Δy\(C7\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C7\}\)\}\(T,x\)=0\.077ϕ1\(y\)\+0\.178ϕ2\(y\)\+0\.003ϕ3\(y\)\+1\.13ϕ4\(y\)−0\.895ϕ5\(y\)\+0\.390ϕ6\(y\),\\displaystyle=0\.077\\,\\phi\_\{1\}^\{\(y\)\}\+0\.178\\,\\phi\_\{2\}^\{\(y\)\}\+0\.003\\,\\phi\_\{3\}^\{\(y\)\}\+1\.13\\,\\phi\_\{4\}^\{\(y\)\}\-0\.895\\,\\phi\_\{5\}^\{\(y\)\}\+0\.390\\,\\phi\_\{6\}^\{\(y\)\},\(35\)Δy\(C9\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C9\}\)\}\(T,x\)=0\.074ϕ1\(y\)−0\.032ϕ2\(y\)−0\.008ϕ3\(y\)\+0\.622ϕ4\(y\)−0\.038ϕ5\(y\)\+0\.315ϕ6\(y\),\\displaystyle=0\.074\\,\\phi\_\{1\}^\{\(y\)\}\-0\.032\\,\\phi\_\{2\}^\{\(y\)\}\-0\.008\\,\\phi\_\{3\}^\{\(y\)\}\+0\.622\\,\\phi\_\{4\}^\{\(y\)\}\-0\.038\\,\\phi\_\{5\}^\{\(y\)\}\+0\.315\\,\\phi\_\{6\}^\{\(y\)\},\(36\)Δy\(C10\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C10\}\)\}\(T,x\)=0\.022ϕ1\(y\)−0\.052ϕ2\(y\)−0\.013ϕ3\(y\)\+0\.259ϕ4\(y\)\+0\.191ϕ5\(y\)\+0\.256ϕ6\(y\),\\displaystyle=0\.022\\,\\phi\_\{1\}^\{\(y\)\}\-0\.052\\,\\phi\_\{2\}^\{\(y\)\}\-0\.013\\,\\phi\_\{3\}^\{\(y\)\}\+0\.259\\,\\phi\_\{4\}^\{\(y\)\}\+0\.191\\,\\phi\_\{5\}^\{\(y\)\}\+0\.256\\,\\phi\_\{6\}^\{\(y\)\},\(37\)Δy\(C12\)\(T,x\)\\displaystyle\\Delta y^\{\(\\mathrm\{C12\}\)\}\(T,x\)=0\.233ϕ1\(y\)\+0\.141ϕ2\(y\)−0\.124ϕ3\(y\)−0\.189ϕ4\(y\)\+0\.448ϕ5\(y\)\+0\.195ϕ6\(y\)\.\\displaystyle=0\.233\\,\\phi\_\{1\}^\{\(y\)\}\+0\.141\\,\\phi\_\{2\}^\{\(y\)\}\-0\.124\\,\\phi\_\{3\}^\{\(y\)\}\-0\.189\\,\\phi\_\{4\}^\{\(y\)\}\+0\.448\\,\\phi\_\{5\}^\{\(y\)\}\+0\.195\\,\\phi\_\{6\}^\{\(y\)\}\.\(38\)
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