Geometry-free prediction of inertial lift forces in microfluidic devices using deep learning

# Geometry-free prediction of inertial lift forces in microfluidic devices using deep learning
Source: [https://arxiv.org/html/2605.08109](https://arxiv.org/html/2605.08109)
Jesse Ward\-BondDepartment of Mechanical & Industrial Engineering, University of TorontoThese authors contributed equally to this work\.Ali MashadianDepartment of Mechanical & Industrial Engineering, University of TorontoMechanical Engineering, Purdue UniversityThese authors contributed equally to this work\.Edmond W\. K\. YoungDepartment of Mechanical & Industrial Engineering, University of TorontoDepartment of Materials Science & Engineering, University of TorontoInstitute of Biomedical Engineering, University of Toronto

###### Abstract

Inertial microfluidic devices \(IMDs\) offer low\-cost, high\-throughput alternative techniques for many traditional particle\- \(or cell\-\) manipulation tasks, but simulating them requires being able to predict particle migration, and thus particle lift forces, under a variety of possible channel geometries\. Recent work has demonstrated that machine learning models can be used to drastically speed up these numerical simulations, but doing so required training individual models for every unique channel cross\-section type \(e\.g\., rectangular, triangular\)—shifting the burden from the simulation step to the training step\. In this paper, we develop a novel approach for predicting particle lift forces that contains no explicit geometric parameters\. We train a neural network model using a new parameter set and show that while it performs comparably to existing models on channel geometries in the training set, it is able to generalize to unseen channel geometries far more effectively\. We show that the lift force model developed herein can be easily transferred to particle tracing simulation software, where it is capable of predicting particle migration patterns consistent with the literature across a variety of channel designs\.

## 1Introduction

Microfluidic technologies have advanced rapidly in the past two decades and are increasingly used for the transport and manipulation of suspended particles within microscale channels\. Among these approaches, inertial microfluidics has emerged as a powerful passive technique for high\-throughput particle and cell separation by exploiting inertial lift forces and secondary flow effects at intermediate Reynolds numbers\.[45](https://arxiv.org/html/2605.08109#bib.bib41)Current applications include cell sorting, circulating tumor cell enrichment, blood fractionation, and size\-based nanoparticle purification\.[42](https://arxiv.org/html/2605.08109#bib.bib14),[21](https://arxiv.org/html/2605.08109#bib.bib29),[46](https://arxiv.org/html/2605.08109#bib.bib7),[16](https://arxiv.org/html/2605.08109#bib.bib22),[43](https://arxiv.org/html/2605.08109#bib.bib23),[12](https://arxiv.org/html/2605.08109#bib.bib24),[30](https://arxiv.org/html/2605.08109#bib.bib25),[44](https://arxiv.org/html/2605.08109#bib.bib18),[4](https://arxiv.org/html/2605.08109#bib.bib20)Future applications are expected to expand toward point\-of\-care diagnostics, continuous bioprocessing, and integrated sample preparation in automated lab\-on\-a\-chip systems\.[19](https://arxiv.org/html/2605.08109#bib.bib42),[18](https://arxiv.org/html/2605.08109#bib.bib43),[39](https://arxiv.org/html/2605.08109#bib.bib44)Inertial microfluidic separation enables label\-free concentration of target particulate species, narrowing of particle size distributions, and continuous operation without external fields or complex actuation, making it particularly attractive for scalable biomedical and industrial workflows\.

Most inertial microfluidic devices \(IMDs\) employ channels with rectangular cross\-sections, largely reflecting legacy fabrication approaches such as soft lithography of poly\(dimethylsiloxane\) \(PDMS\), dry etching of glass, and micromilling of thermoplastics\.[34](https://arxiv.org/html/2605.08109#bib.bib46),[13](https://arxiv.org/html/2605.08109#bib.bib45),[17](https://arxiv.org/html/2605.08109#bib.bib47)These fabrication methods naturally produce vertical sidewalls and planar geometries, which have historically constrained channel design despite the strong dependence of inertial focusing behaviour on channel cross\-sectional shape\. Significant research effort has therefore focused on optimizing flow conditions within rectangular geometries, as well as minimizing fabrication artifacts such as unintended sidewall bevelling that produces trapezoidal cross\-sections and modifies equilibrium particle focusing positions\. Alternative low\-cost fabrication approaches, including thermally shrunk polymer methods \(“Shrinky Dink”\), have enabled semi\-circular channel geometries; however, these techniques have generally suffered from limited resolution, poor surface smoothness, and reduced reproducibility, restricting their adoption for precision inertial flow studies\.[11](https://arxiv.org/html/2605.08109#bib.bib48)

Recent advances in high\-resolution additive manufacturing are beginning to transform microfluidic fabrication by enabling complex three\-dimensional architectures that were previously inaccessible using conventional methods\. Modern 3D printing technologies now permit fabrication of non\-rectangular cross\-sections with improved surface finish, optical transparency, and microscale feature resolution\. Because inertial particle migration arises from the interplay between shear\-gradient lift forces, wall\-induced lift forces, and geometry\-dependent secondary flows, channel cross\-sectional shape represents a critical yet underexplored design parameter\. Leveraging emerging fabrication capabilities provides an opportunity to move beyond traditional rectangular channels, expand the geometric design space, and engineer flow fields that enhance particle focusing stability, separation resolution, and throughput\.[38](https://arxiv.org/html/2605.08109#bib.bib49)Such advances may enable the next generation of IMDs optimized through geometry\-driven control of particle dynamics\.

![Five-panel flowchart: (a) 3D renders of rectangular, triangular, semicircular, and trapezoidal channels; (b) a heat map of velocity and a vector field of ground-truth lift forces, labeled with nondimensional parameters; (c) a four-layer FCNN architecture diagram followed by two scatter plots comparing DNS results to NN predictions for lift force components; (d) a software pipeline showing the model moving from PyTorch through ONNX to MATLAB and COMSOL; (e) 3D visualizations of particle focusing in square, cylindrical, and serpentine channels.](https://arxiv.org/html/2605.08109v1/figure-abstract.png)Figure 1:Overview of development and downstream integration of a geometry\-free lift force model\. \(a\) Direct numerical simulations \(DNSs\) are conducted to generate velocity profiles and lift force maps\. \(b\) Results of the simulations are transformed according to our parametrization and augmented to be used as training data\. \(c\) A fully connected neural network \(FCNN\) is trained to predict the x\- and y\-components of the lift force\. \(d\) The model is exported for use in simulation software\. \(e\) Particle\-tracing simulations are conducted using model\-predicted lift forces\.Fundamentally, the tendency of suspended particles flowing through a conduit to migrate to certain equilibrium positions, which was first observed experimentally at the macroscale bySegré and Silberberg[32](https://arxiv.org/html/2605.08109#bib.bib6)and then later observed byDi Carloet al\.[9](https://arxiv.org/html/2605.08109#bib.bib2)at the microscale, depends on the net lift force acting on each particle\. Despite advances in both theory and modelling technologies, it remains challenging to predict the net lift force in a computationally cheap and geometrically flexible manner\.

Researchers have attempted to derive analytical expressions for the lift forces acting on particles in IMDs, mostly through asymptotic analysis\.[4](https://arxiv.org/html/2605.08109#bib.bib20)However, these approaches rely on the limiting assumptions that particle diameters and Reynolds numbers \(R​eRe\) are asymptotically small\. Furthermore, these expressions are developed for particular channel geometries, and are not always readily adaptable\. Although much work has been done to extend analytical approaches to more realistic microfluidic conditions,[15](https://arxiv.org/html/2605.08109#bib.bib3),[2](https://arxiv.org/html/2605.08109#bib.bib15),[36](https://arxiv.org/html/2605.08109#bib.bib16)these limitations persist\.

Given the challenges with analytical and semi\-analytical approaches, IMD design typically relies on either physical design iterations or direct numerical simulations \(DNSs\)\. The former approach involves designing and fabricating the device, testing it on the bench, and repeating this process by trial and error until an optimal combination of design and operating characteristics emerges\. This slow and stochastic process has spurred research into numerical simulation techniques that can predict the inertial migration of particlesin silico\.[8](https://arxiv.org/html/2605.08109#bib.bib1),[22](https://arxiv.org/html/2605.08109#bib.bib21),[23](https://arxiv.org/html/2605.08109#bib.bib9),[33](https://arxiv.org/html/2605.08109#bib.bib4)One common simulation workflow involves generating a map of lift forces by repeatedly fixing a stationary particle at different locations within a channel cross\-section, allowing it to freely rotate and interact with the fluid, and then solving the underlying force balance to determine the net lift force\. Once the lift force map is obtained, Lagrangian particle\-tracing simulations can be used to determine particle migration\.[23](https://arxiv.org/html/2605.08109#bib.bib9),[33](https://arxiv.org/html/2605.08109#bib.bib4)Although simulation results have shown great agreement with experimental results, they are computationally intensive and need to be re\-run when design conditions change, making this workflow far from optimal\.

Advances in machine learning \(ML\) have been steadily percolating into microfluidic design workflows,[25](https://arxiv.org/html/2605.08109#bib.bib26),[29](https://arxiv.org/html/2605.08109#bib.bib27)and some attempts have been made at accelerating IMD design with ML\. InSuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5), the authors trained multiple fully connected neural networks \(FCNN\) to generate lift force maps for straight channels with rectangular, semicircular, or triangular cross\-sections, and then demonstrated that these maps closely agree with results from both physical experiments and DNSs\.Hamdiet al\.[14](https://arxiv.org/html/2605.08109#bib.bib50)later extended that work by combining the neural networks for each cross section into a single neural network where cross\-section shape was represented with a one\-hot encoded vector and used as an input to their neural network\.Safariet al\.[31](https://arxiv.org/html/2605.08109#bib.bib28)andAkhbari and Nasr Esfahany[1](https://arxiv.org/html/2605.08109#bib.bib19)trained FCNN and log\-log regression models, respectively, to predict final particle positions in spiral channels, and were able to obtain lift coefficients and particle positions consistent with literature\.[31](https://arxiv.org/html/2605.08109#bib.bib28),[1](https://arxiv.org/html/2605.08109#bib.bib19)These approaches demonstrate the feasibility of ML\-assisted particle migration prediction, but have all resulted in models with explicit geometric dependence: they are either developed for particular channel cross\-section shapes, or depend on parameters such as the channel’s radius of curvature, which are not defined for many types of IMDs\.

In this paper, we introduce the first geometry\-agnostic ML model for prediction of lift forces of particles travelling in microfluidic channels\. We develop a parametrization scheme that allows this prediction to be independent of references to channel geometry and particle location\. We show that the models developed using our parametrization scheme can generalize across diverse channel geometries that were not present in the training data\. We further demonstrate that these models can be easily integrated intoCOMSOLand used to accurately predict particle migration in IMDs\.

## 2Methods

[Figure˜1](https://arxiv.org/html/2605.08109#S1.F1)gives an overview of the methodology used in this study\. First, DNSs were conducted with and without particle tracing to obtain “undisturbed” \(particle\-free\) velocity profiles and lift force maps for four different channel cross\-sectional shapes under various flow conditions \([Fig\.˜1](https://arxiv.org/html/2605.08109#S1.F1)a\)\. This data was augmented \([Fig\.˜1](https://arxiv.org/html/2605.08109#S1.F1)b\) and a portion of this data was used to train and validate a FCNN model \([Fig\.˜1](https://arxiv.org/html/2605.08109#S1.F1)c\)\. The trained model was then exported toCOMSOLMultiPhysics \([Fig\.˜1](https://arxiv.org/html/2605.08109#S1.F1)d\) and used to simulate different IMDs \([Fig\.˜1](https://arxiv.org/html/2605.08109#S1.F1)e\)\. Note that for straight channels, we used the convention that channel cross\-sections are in thex​yxy\-plane, and that the channels extend longitudinally in thezz\-direction\.

### 2\.1Geometry\-agnostic formulation

In this section, we first identify the geometry\-dependent parameters in a common lift force formulation\. We then replace these parameters directly with the local velocity profile around the particle, which we model using a second\-order Taylor series expansion\. Finally, we non\-dimensionalize our set of geometry\-free parameters and use these as inputs to an FCNN\.

The lift force,𝐅L\\mathbf\{F\}\_\{\\texttt\{L\}\}, on a neutrally buoyant particle in a straight rectangular channel is commonly formulated as

𝐅L=f​\(aH,x0H,y0H,A​R,R​ec\),\\mathbf\{F\}\_\{\\texttt\{L\}\}=f\\left\(\\frac\{a\}\{H\},\\frac\{x\_\{0\}\}\{H\},\\frac\{y\_\{0\}\}\{H\},AR,Re\_\{c\}\\right\),\(1\)whereaais the particle diameter,HHis the channel height,\(x0,y0\)\(x\_\{0\},y\_\{0\}\)is the position of the center of the particle within the channel cross\-section,A​RARis the aspect ratio of the channel, andR​ecRe\_\{c\}is the channel Reynolds number\.[23](https://arxiv.org/html/2605.08109#bib.bib9),[35](https://arxiv.org/html/2605.08109#bib.bib5)\.

Models developed from formulations like[Eq\.˜1](https://arxiv.org/html/2605.08109#S2.E1)face two major challenges:\(a\)they depend on geometric characteristics that may not be defined for every channel type \(e\.g\., aspect ratio is not well\-defined for a trapezoid\), and\(b\)downstream simulations using[Eq\.˜1](https://arxiv.org/html/2605.08109#S2.E1)will need to constantly determine channel cross\-section geometries and the relative position of particles within those geometries, which may not be feasible in all channel geometries \(e\.g\., expansion\-contraction channels,[21](https://arxiv.org/html/2605.08109#bib.bib29)or serpentine channels[44](https://arxiv.org/html/2605.08109#bib.bib18)\)\.

To address these challenges, we developed a novel model parametrization that does not explicitly depend on channel geometries\. To do this, we first note that these geometric terms are only necessary insofar as they ultimately help specify the local velocity field, and thus the fluid\-induced stresses, around a particle at a given position in the channel cross\-section\. We further note the following transitive relationships: the inertial lift force acting on a particle is determined by the fluid\-induced stresses, which is in turn determined by the local velocity field, which is ultimately a function of the shape of the undisturbed velocity profile and the fluid and particle characteristics \([Fig\.˜2](https://arxiv.org/html/2605.08109#S2.F2)\)\. We exploit these transitive relationships to model the lift force acting on a particle as:

𝐅L=f​\(ρ,μ,a,𝒰\)\\mathbf\{F\}\_\{\\texttt\{L\}\}=f\(\\rho,\\mu,a,\\mathcal\{U\}\)\(2\)whereρ\\rhois the fluid density,μ\\muis the dynamic viscosity, and

𝒰:=\{𝐮​\(𝐱\)\|‖𝐱−𝐱𝟎‖2=a2\}\\mathcal\{U\}:=\\left\\\{\\mathbf\{u\}\(\\mathbf\{x\}\)\\;\\middle\|\\;\\\|\\mathbf\{x\}\-\\mathbf\{x\_\{0\}\}\\\|\_\{2\}=\\frac\{a\}\{2\}\\right\\\}\(3\)is the set of velocity vectors𝐮\\mathbf\{u\}at every point on the surface of the particle, centred at𝐱𝟎\\mathbf\{x\_\{0\}\}, in the absence of fluid\-particle interactions \(i\.e\., using the undisturbed velocity profile\)\.

![A schematic diagram illustrating a four-step transitive process: (1) An undisturbed velocity profile in a rectangular channel with laminar streamlines and a red parabolic vector overlay; (2) plus a spherical particle; (3) results in a disturbed velocity profile showing streamlines curving around the sphere and a green wake of secondary flow; (4) which corresponds to a spherical heat map of fluid stress on the particle surface, terminating in an arrow pointing to the lift force variable F_L.](https://arxiv.org/html/2605.08109v1/transitive.png)Figure 2:The transitive relationship between the undisturbed velocity profile and the lift force𝐅L\\mathbf\{F\}\_\{\\texttt\{L\}\}acting on a particle\.Models are only trained in straight channels containing Poiseuille flow, and thus velocity vectors in the undisturbed flow are only in thezz\-direction \(i\.e\., the flow is perfectly perpendicular to the cross\-section\)\. Furthermore, the inertial lift force is zero in thezz\-direction\. We can therefore simplify[Eq\.˜2](https://arxiv.org/html/2605.08109#S2.E2)to the two\-dimensional case, only considering the velocities in thezz\-direction:

𝐅LX​Y≈f​\(ρ,μ,a,𝒲\)\\mathbf\{F\}\_\{\\texttt\{L\}\_\{XY\}\}\\approx f\(\\rho,\\mu,a,\\mathcal\{W\}\)\\\\\(4\)wherew​\(x,y\)w\(x,y\)is thezz\-direction velocity at point\(x,y\)\(x,y\)on the surface of the imaginary particle and

𝒲:=\{w​\(x,y\)\|\(x−x0\)2\+\(y−y0\)2=\(a2\)2\}\\mathcal\{W\}:=\\left\\\{w\(x,y\)\\;\\middle\|\\;\(x\-x\_\{0\}\)^\{2\}\+\(y\-y\_\{0\}\)^\{2\}=\\left\(\\frac\{a\}\{2\}\\right\)^\{2\}\\right\\\}\(5\)is the set of all velocities in thezz\-direction,ww, around the circumference of that particle in thex​yxy\-plane\.

Most particle\-tracing simulations treat particles as point masses, and thus𝒲\\mathcal\{W\}is not computable\. Instead,𝒲\\mathcal\{W\}can be approximated with the second\-order Taylor series expansion around the particle centre \(see Supplementary Information\), which leads to the formulation:

𝐅LX​Y≈f​\(ρ,μ,a,w,wx,wy,wx​x,wy​y,wx​y\)\\mathbf\{F\}\_\{\\texttt\{L\}\_\{XY\}\}\\approx f\(\\rho,\\mu,a,w,w\_\{x\},w\_\{y\},w\_\{xx\},w\_\{yy\},w\_\{xy\}\)\(6\)wherewxw\_\{x\}andwyw\_\{y\}are first\-order partial derivatives of the velocity field at the centre of the particle, andwx​xw\_\{xx\},wx​yw\_\{xy\}andwy​yw\_\{yy\}are the second\-order derivatives\.

With the reformulation in[Eq\.˜6](https://arxiv.org/html/2605.08109#S2.E6), there is no longer any explicit dependence on channel geometries, and all model inputs are computationally tractable\. All that remains is to non\-dimensionalize the formulation using the Buckinghamπ\\pitheorem\.[6](https://arxiv.org/html/2605.08109#bib.bib12)This results in a final expression for the dimensionless lift coefficient,𝐂LX​Y\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XY\}\}, given as:

𝐂LX​Y=𝐅LX​Yρ​w2​a2=f​\(ρ​w​aμ,wx​aw,wy​aw,wx​x​a2w,wy​y​a2w,wx​y​a2w\)\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XY\}\}=\\frac\{\\mathbf\{F\}\_\{\\texttt\{L\}\_\{XY\}\}\}\{\\rho w^\{2\}a^\{2\}\}=f\\left\(\\frac\{\\rho wa\}\{\\mu\},\\frac\{w\_\{x\}a\}\{w\},\\frac\{w\_\{y\}a\}\{w\},\\frac\{w\_\{xx\}a^\{2\}\}\{w\},\\frac\{w\_\{yy\}a^\{2\}\}\{w\},\\frac\{w\_\{xy\}a^\{2\}\}\{w\}\\right\)\(7\)For convenience, we use the following shorthand to represent model[Eq\.˜7](https://arxiv.org/html/2605.08109#S2.E7)going forward:

𝐂L=f​\(R​ep,w¯x,w¯y,w¯x​x,w¯y​y,w¯x​y\)\.\\mathbf\{C\}\_\{\\texttt\{L\}\}=f\(Re\_\{p\},\\bar\{w\}\_\{x\},\\bar\{w\}\_\{y\},\\bar\{w\}\_\{xx\},\\bar\{w\}\_\{yy\},\\bar\{w\}\_\{xy\}\)\.\(8\)where the overbar indicates non\-dimensionalized parameters\. Note that we only consider lift force in thex​yxy\-plane for the remainder of this study\.

### 2\.2Data generation

Lift force maps for particles in rectangular \(ℛ\\mathcal\{R\}\), triangular \(𝒯\\mathcal\{T\}\), and semicircular \(𝒮\\mathcal\{S\}\) channels were taken from existing datasets reported in literature\. These datasets contain lift force maps for different particle sizes, maximum fluid velocities \(UmU\_\{m\}\) and, for the rectangular channels only, aspect ratios, as outlined in[Fig\.˜3](https://arxiv.org/html/2605.08109#S2.F3)\.[35](https://arxiv.org/html/2605.08109#bib.bib5)An additional independently\-generated trapezoidal dataset was created following the process in Shamlooet al\.[33](https://arxiv.org/html/2605.08109#bib.bib4)Datasets were augmented by rotating in\\qty20 increments and flipping about theXX\-axes, resulting in a 36\-fold increase in the size of each dataset\. Each dataset was split into training/validation/testing sets \(0\.7/0\.15/0\.15\) denotedℛt​r\\mathcal\{R\}^\{tr\},ℛv\\mathcal\{R\}^\{v\},ℛt​s\\mathcal\{R\}^\{ts\}, respectively\.

![Four geometric cross-sections plotted on xy-coordinate planes with vertices labeled: (1) a red square with a black star at the origin (0,0) and a vertex at (25,25), featuring dashed lines along the axes; (2) a green right triangle with vertices at (0,-25), (25,-25), and (0,25), with a vertical dashed line and a star on the y-axis; (3) a blue quadrant (quarter-circle) with vertices at (0,-25) and (50,-25), a dashed vertical line, and a star on the y-axis; (4) a purple trapezoid with vertices at (-300,0), (300,0), (300,150), and (-300,80), with a star located at the midpoint of the base. An xyz coordinate legend indicates x is horizontal, y is vertical, and z points into the page.](https://arxiv.org/html/2605.08109v1/cross-sections-origin.png)
GeometryARa\(\\unit\\micro\)UmU\_\{m\}\(\\unit\\per\)SizeSourceℛ\\mathcal\{R\}\{1,2,3,4\}\\\{1,2,3,4\\\}\{5\.0,7\.5,10\.0,12\.5,15\.0\}\\\{5\.0,7\.5,10\.0,12\.5,15\.0\\\}\{1\.0,2\.0,3\.0,4\.0\}\\\{1\.0,2\.0,3\.0,4\.0\\\}8068[35](https://arxiv.org/html/2605.08109#bib.bib5)𝒯\\mathcal\{T\}\-\{5\.0,7\.5,10\.0,12\.5,15\.0\}\\\{5\.0,7\.5,10\.0,12\.5,15\.0\\\}\{1\.0,2\.0,3\.0,4\.0\}\\\{1\.0,2\.0,3\.0,4\.0\\\}2420[35](https://arxiv.org/html/2605.08109#bib.bib5)𝒮\\mathcal\{S\}\-\{5\.0,7\.5,10\.0,12\.5,15\.0\}\\\{5\.0,7\.5,10\.0,12\.5,15\.0\\\}\{1\.0,2\.0,3\.0,4\.0\}\\\{1\.0,2\.0,3\.0,4\.0\\\}3672[35](https://arxiv.org/html/2605.08109#bib.bib5)𝒫\\mathcal\{P\}\-\{10\.0,12\.5,25\.0\}\\\{10\.0,12\.5,25\.0\\\}\{0\.25,0\.52,0\.63,1\.3,2\.0,2\.6,4\.2\}\\\{0\.25,0\.52,0\.63,1\.3,2\.0,2\.6,4\.2\\\}1408This paper

Figure 3:Summary of raw datasets used for model training and testing\. From left to right: rectangular \(ℛ\\mathcal\{R\}\), triangular \(𝒯\\mathcal\{T\}\), semicircular \(𝒮\\mathcal\{S\}\), and trapezoidal \(𝒫\\mathcal\{P\}\)\. Solid lines indicate channel walls and dashed lines are lines of symmetry and also indicate the geometric limits of the simulated data\. Table abbreviations: aspect ratio \(AR\), particle size \(a\), maximum flow velocities \(UmU\_\{m\}\), cardinality \(size\), and source of each dataset\. Note that we did not use all combinations of particle sizes and maximum flow velocities for the trapezoidal cross\-sections\.
### 2\.3Machine learning model

A fully connected neural network \(FCNN\) was used to approximate[Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8)\. In these networks the input to a given neuron is a linear combination of the outputs from all neurons in the previous layer\. Inputs were the arguments of[Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8)and outputs were the𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}components:\(CLx,CLy\)\(C\_\{\\texttt\{L\}\_\{x\}\},C\_\{\\texttt\{L\}\_\{y\}\}\)\. The ReLU activation function was used for all layers except the final layer, which used linear activation functions\. Models were trained using minibatch gradient descent with Nesterov Momentum for 300 epochs with early stopping if the validation loss did not improve over the incumbent after 30 epochs\.[27](https://arxiv.org/html/2605.08109#bib.bib30),[37](https://arxiv.org/html/2605.08109#bib.bib10)The loss function was

ℒ​\(ℬt\)=1\|ℬt\|​∑i=1\|ℬt\|‖𝐂Li−𝐂^Li‖22\\mathcal\{L\}\(\\mathcal\{B\}\_\{t\}\)=\\frac\{1\}\{\\left\|\\mathcal\{B\}\_\{t\}\\right\|\}\\sum\_\{i=1\}^\{\\left\|\\mathcal\{B\}\_\{t\}\\right\|\}\\left\\\|\\mathbf\{C\}\_\{\\texttt\{L\}\}^\{i\}\-\\hat\{\\mathbf\{C\}\}\_\{\\texttt\{L\}\}^\{i\}\\right\\\|\_\{2\}^\{2\}\(9\)whereℬt\\mathcal\{B\}\_\{t\}is batchttandC^Lxi\\hat\{C\}\_\{\\texttt\{L\}\_\{x\}\}^\{i\}is the predictedxx\-component of the lift coefficient vector for theit​hi^\{th\}data point in batchℬt\\mathcal\{B\}\_\{t\}\. Hyperparameter tuning was done on the validation datasets following the methodology outlined inGodboleet al\.[10](https://arxiv.org/html/2605.08109#bib.bib17)The final architecture was6×256×128×64×26\\times 256\\times 128\\times 64\\times 2\([Fig\.˜4](https://arxiv.org/html/2605.08109#S2.F4)\)\.

![A diagram of a neural network with four hidden layers.](https://arxiv.org/html/2605.08109v1/fcnn.png)Figure 4:Model architecture\. Input nodes are on the left, output nodes on the right\. Hidden layers are represented in blue\. Biases \(b\) are outlined in green\.NNis the number of neurons per layer\.ψ\\psiare model weights\.In addition to mean squared error \(MSE\), we also evaluate models using the angular error of predictions, which we define as

ϕe​r​r=cos−1⁡\(𝐂L⋅𝐂L^∥𝐂L∥⋅∥𝐂L^∥\),\\phi^\{err\}=\\cos^\{\-1\}\\left\(\\frac\{\\mathbf\{C\}\_\{\\texttt\{L\}\}\\cdot\\hat\{\\mathbf\{C\}\_\{\\texttt\{L\}\}\}\}\{\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert\\cdot\\lVert\\hat\{\\mathbf\{C\}\_\{\\texttt\{L\}\}\}\\rVert\}\\right\),\(10\)and the relative magnitude error of predictions, which we define as

∥𝐂L∥e​r​r=\|∥𝐂L^∥−∥𝐂L∥∥𝐂L∥\|×100%\.\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}=\\lvert\\frac\{\\lVert\\hat\{\\mathbf\{C\}\_\{\\texttt\{L\}\}\}\\rVert\-\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert\}\{\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert\}\\rvert\\times 100\\%\.\(11\)Collectively, these two metrics provide a measure of the accuracy of thedirectionand themagnitudeof the predicted lift force, respectively\.

A baseline model was trained onℛt​r\\mathcal\{R\}^\{tr\}inMATLABaccording to the methodology developed bySuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5)This training used the same dataset, architecture, Levenberg–Marquardt loss function, and stopping conditions as reported by the original authors\. To compare that model to our model, both were trained and tested on the same unrotated, unflipped, rectangular datasetℛ0⊂ℛ\\mathcal\{R\}\_\{0\}\\subset\\mathcal\{R\}, which contained 5660 training points and 2328 testing points, a 70/30 split\. To enable direct comparison, outputs from the baseline model were converted to match our𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}formulation \(see Supplementary Information\)\.

Feature importance was estimated using Shapley values\. Shapley values were in turn estimated using theKernelExplainermodule from theSHAPlibrary on 500 samples fromℛt​s\\mathcal\{R\}^\{ts\}\.[24](https://arxiv.org/html/2605.08109#bib.bib13)

### 2\.4Downstream simulation

Trained models were converted into the Open Neural Network Exchange \(ONNX\) format\.[3](https://arxiv.org/html/2605.08109#bib.bib31)This format allowed the model to be imported intoMATLAB, which in turn enabled it to be used for particle\-tracing simulations withinCOMSOLMultiphysics\. Particle tracing simulations were performed for both straight and curved channels\. Straight channel geometries included channels with square, triangular, circular, trapezoidal, and rhombic cross\-sections, as well as expansion\-contraction channels with rectangular cross\-sections\. Curved channel geometries included the standard spiral and u\-turn serpentine designs, as well as a “reverse wavy” serpentine geometry based on the work byZhouet al\.[47](https://arxiv.org/html/2605.08109#bib.bib32)

All the training data used in this paper were from straight microchannels, so the model was parametrized only with velocity gradients in theX​YXY\-direction \([Eq\.˜7](https://arxiv.org/html/2605.08109#S2.E7)\)\. We used a simple rotational mapping to account for the non\-zeroZZ\-gradients in curved channels\. Where necessary, velocity gradients in the three dimensionalX​Y​ZXYZ\-space were converted to theX​YXY\-plane with a rotational mapping that aligned the velocity vector at the particle centre with theZZ\-direction\. This mapping was obtained using Rodrigues’ rotation formula\.[7](https://arxiv.org/html/2605.08109#bib.bib33)The predicted𝐂LX​Y\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XY\}\}values were then transformed back into the correct direction using the inverse mapping\. Full details can be found in Supplementary Information\.

## 3Results

### 3\.1Model performance

[Figure˜5](https://arxiv.org/html/2605.08109#S3.F5)shows the prediction performance of a model trained with our parametrization compared to one trained with the parametrization fromSuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5)When both models are trained and tested on unrotated, unflipped datasets, we found that our model had a lower MSE than the model fromSuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5)\. Despite this lower error, our model was slightly worse at predicting both the magnitude and direction of𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}vectors\. The median angle and magnitude errors for our model were\\qty2\.9 and\\qty6\.4 respectively, compared to\\qty2\.2 and\\qty4\.8 for the re\-implemented Su model \([Fig\.˜5](https://arxiv.org/html/2605.08109#S3.F5), inset\)\. Percentile plots confirmed that the re\-implemented model predicted magnitude and angle better for up to\\qty70 and\\qty95 of samples, respectively \([Fig\.˜5](https://arxiv.org/html/2605.08109#S3.F5)\)\.

![Two side-by-side percentile plots comparing the Current Study (solid blue line) to Su et al. (dashed orange line). Both plots feature a logarithmic y-axis and a linear x-axis from 0 to 100 percentile. The left plot shows relative magnitude error in percent, with both models tracking closely between 0.1% and 100% for most of the distribution. The right plot shows angular error in degrees, with both curves following a similar upward trend between 0.001 and 1 degree. In both plots, the dashed orange line remains slightly below the solid blue line across the middle percentiles, indicating slightly lower errors for the model from Su et al. before both converge at the extreme high percentiles.](https://arxiv.org/html/2605.08109v1/quantile_plot.png)

MetricCurrent StudySuet al\.MSE8\.64×10−78\.64\\text\{\\times\}\{10\}^\{\-7\}9\.93×10−79\.93\\text\{\\times\}\{10\}^\{\-7\}ϕ50e​r​r\\mathbf\{\\phi\}^\{err\}\_\{50\}\(\\unit\)2\.92\.92\.22\.2∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}\(%\)6\.46\.44\.84\.8𝑹𝒙𝟐R^\{2\}\_\{x\}0\.9950\.9950\.9950\.995𝑹𝒚𝟐R^\{2\}\_\{y\}0\.9950\.9950\.9930\.993

Figure 5:Comparison between our model and the model developed inSuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5)Top: percentile plots of the relative magnitude error \(left\) and angle \(right\) between actual and predicted lift force coefficients\. Bottom: summary statistics\.ϕ50e​r​r\\mathbf\{\\phi\}^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}are the median angle and relative magnitude errors, respectively\.To investigate the generalization performance of our model, we re\-trained it several times with increasing amounts of data from new geometries \([Fig\.˜6](https://arxiv.org/html/2605.08109#S3.F6)\)\. When trained only on rectangular data, the model \(predictably\) performed best on the rectangular test set:\\qty50 of predicted vectors are within\\qty4\.6 of their true directions, and within\\qty8\.7 of their true magnitude\. When additional geometries were added to the training data, the model performed better on the corresponding test set for the new geometry, at the cost of some performance loss on the existing geometries\. For example, when triangular training data was added, the median angle and magnitude errors \(ϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}\) on the triangular test data were\\qty57 and\\qty69 lower than they were when the model was only trained on rectangular data\. At the same time, adding this data caused theϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}scores to increase for the rectangular test data by\\qty0\.1 and\\qty0\.8 respectively\. There was a similar pattern when data from semicircular geometries were added to the training data, although in this case the increase inϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}scores across the other geometries was not as pronounced\. Adding the trapezoidal training data resulted in the largest increase inϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}scores across all geometries, likely because this data contains flow rates and particle sizes that the other datasets do not\.

![Three-panel stacked plot showing error metrics for different test sets, grouped by color: Rectangular (red), Triangular (green), Semicircular (blue), and Trapezoidal (purple). Within each color group, four bars or boxplots represent training sets of increasing complexity, distinguished by textures: solid (Rectangular only), diagonal stripes (Rectangular and Triangular), cross-hatch (Rectangular, Triangular, and Semicircular), and dots (all four geometries). The top panel shows Mean Squared Error (MSE) on a logarithmic scale. Adding a geometry to the training set significantly drops its MSE in the corresponding test group. The middle panel shows boxplots for angular error in degrees, and the bottom panel shows boxplots for relative magnitude error in percent. Across all panels, adding geometries to the training set increases test performance on that same geometry, at the cost of decreasing test performance on existing geometries in the training dataset. The Trapezoidal test set (purple) maintains relatively high and consistent error across all training configurations compared to the other geometries](https://arxiv.org/html/2605.08109v1/generalization_boxplots.png)Figure 6:Model performance on seen and unseen geometries as new geometries are added to the training data\. Models were retrained completely when additional geometries were added to the training set\.ℛ,𝒯,𝒮,𝒫\\mathcal\{R,T,S,P\}, represent the rectangular, triangular, semicircular, and trapezoidal datasets, respectively\. Superscriptst​rtrandt​stsrepresent training and testing subsets\.Across all the geometries, and for all permutations of training data tested in[Fig\.˜6](https://arxiv.org/html/2605.08109#S3.F6), theϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}errors were highest for the semicircular test data\. The90t​h90^\{th\}percentile angle and magnitude errors \(ϕ90e​r​r\\phi^\{err\}\_\{90\}and∥𝐂L∥90e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{90\}\) were highest in the trapezoidal data\. This suggests that the trapezoidal test data has a wider distribution, which we confirmed with principal component analysis \(see Supplementary Information\)\.

[Figure˜7](https://arxiv.org/html/2605.08109#S3.F7)shows a qualitative examination of the ability of our model to generalize to triangular, semicircular, and trapezoidal geometries when only trained on rectangular data\. The model was able to capture much of the complex𝐂Lx\\mathbf\{C\}\_\{\\texttt\{L\}\_\{x\}\}and𝐂Ly\\mathbf\{C\}\_\{\\texttt\{L\}\_\{y\}\}variation within the cross sections, although it was slightly less faithful at capturing𝐂Lx\\mathbf\{C\}\_\{\\texttt\{L\}\_\{x\}\}variation across theXX\-direction than𝐂Ly\\mathbf\{C\}\_\{\\texttt\{L\}\_\{y\}\}variation across theYY\-direction\. These results were consistent across all flow rates and particle sizes in[Fig\.˜3](https://arxiv.org/html/2605.08109#S2.F3)\.

![Six rows of comparison plots for three channel geometries: triangular (top two rows), semicircular (middle two rows), and trapezoidal (bottom two rows). Each row consists of three panels: a heatmap of actual values, a heatmap of predicted values, and a scatter plot of a meridian cross-section. The heatmaps use a diverging blue-to-red color scale centered at zero. The first and second columns show that the neural network predictions qualitatively capture the spatial distribution of the CL_x and CL_y components across all three geometries. The third column contains scatter plots titled "Horiz. Meridian" or "Vert. Meridian," where blue dots (actual) and pink crosses (predicted) are plotted against coordinate position. These plots show close agreement in trends, though the predicted values exhibit more visible oscillation and noise for the CL_x values in the triangular and semicircular cases compared to the other figures. Insets in the scatter plots indicate the specific horizontal or vertical meridian line used to create the final figure in each row.](https://arxiv.org/html/2605.08109v1/heatmaps.png)Figure 7:Comparison of DNS\- and FCNN\-predictedCLxC\_\{\\texttt\{L\}\_\{x\}\}andCLyC\_\{\\texttt\{L\}\_\{y\}\}profiles in straight channels with different cross\-sections\. The model was trained only on rectangular cross sections\. Channel dimensions are not to scale\.
### 3\.2Downstream simulation

Straight channel simulations included square, triangular, circular, trapezoidal, and rhomboidal cross\-sections\.[Figure˜8](https://arxiv.org/html/2605.08109#S3.F8)a and[Fig\.˜8](https://arxiv.org/html/2605.08109#S3.F8)b show the results for rectangular and trapezoidal cross\-sections\. In all cases, the simulated particles migrated to focal points matching those reported in the literature\. For example,[Fig\.˜8](https://arxiv.org/html/2605.08109#S3.F8)a illustrates a particle\-tracing simulation in a\\qty100\\micro×\\times\\qty100\\micro×\\times\\qty5\\centirectangular channel\. Particles of varying sizes were released uniformly at the inlet and migrated to the four equilibrium positions observed in the literature\. Similarly,[Fig\.˜8](https://arxiv.org/html/2605.08109#S3.F8)b shows\\qty10\\microparticles focused to the expected three positions in a triangular channel with a\\qty100\\microbase\. We additionally obtained particle focusing results consistent with the literature for circular, trapezoidal, and rhombic cross\-sections \([Section˜S7](https://arxiv.org/html/2605.08109#Ax1.SS7)\)\.

![Four-panel simulation summary: (a) Particle tracing in a straight square channel where red, green, and blue particles of different sizes transition from a uniform grid at the inlet to four focused equilibrium positions at the outlet, shown with a corresponding lift force vector map; (b) tracing in a triangular channel where yellow particles migrate from a random distribution to three equilibrium positions near the midpoints of the channel walls; (c) An expansion-contraction simulation with two particle sizes. After 25 chambers, the smaller particles have migrated to two streamlines closer to the channel wall, while the large particles have stayed in the middle. This is presented alongside experimental results from literature validating this migration pattern. (d) As in (c), but in a serpentine channel. The particles exhibit the same migration pattern.](https://arxiv.org/html/2605.08109v1/simulation-figures-2.png)Figure 8:Particle trajectories fromCOMSOLsimulations employing our lift\-force model trained on straight channels with rectangular, semicircular, triangular, and trapezoidal cross\-sections\. \(a\) A straight square channel containing\\qty10\\micro\(blue\),\\qty15\\micro\(green\), and\\qty20\\micro\(red\) particles\. \(b\) A straight triangular channel containing\\qty10\\microparticles\. \(c\) Twenty\-five chambers of an expansion\-contraction channel containing 5\.5 \(red\) and\\qty9\.9\\microparticles\. \(d\) A 15\-unit u\-turn serpentine channel containing\\qty3\\micro\(red\) and\\qty10\\micro\(green\) particles\. Experimental results from literature for \(c\) were adapted fromWuet al\.[41](https://arxiv.org/html/2605.08109#bib.bib34)with permission from the Royal Society of Chemistry\. Experimental results from literature for \(d\) were adapted fromZhanget al\.[44](https://arxiv.org/html/2605.08109#bib.bib18)with permission from Springer Nature\.We further tested the performance of our model in particle tracing simulations in curved channels, channels with non\-constant cross sections, and channels with turns\.[Figure˜8](https://arxiv.org/html/2605.08109#S3.F8)c shows the results of simulating particle tracing for\\qty5\.5\\microand\\qty9\.9\\microparticles flowing through an expansion\-contraction channel\. Our simulation predicted that, after 25 chambers, the smaller particles fully migrate to streamlines closer to the channel walls, while the larger particles focus on the channel centerline\. These simulation results are closely aligned with experimental findings in existing literature\.[41](https://arxiv.org/html/2605.08109#bib.bib34)[Figure˜8](https://arxiv.org/html/2605.08109#S3.F8)d shows similar agreement between a simulated serpentine channel using our lift force model and experimental observations from literature: the smaller particles \(\\qty3\\micro, red\) migrated to streamlines closer to channel walls, while the larger particles \(\\qty10\\micro, green\) focused along the channel centerline\.[44](https://arxiv.org/html/2605.08109#bib.bib18)Additional comparisons between simulations and experimental observations for “reverse\-wavy” and spiral IMDs are available in the Supplementary Information\.

### 3\.3Feature importance

Feature importance analysis showed thatw¯x\\bar\{w\}\_\{x\}andw¯y\\bar\{w\}\_\{y\}had the greatest impact onCLxC\_\{\\texttt\{L\}\_\{x\}\}andCLyC\_\{\\texttt\{L\}\_\{y\}\}, respectively\. As expected from the shape of the velocity profile and the nature of wall\-induced lift forces \([Fig\.˜9](https://arxiv.org/html/2605.08109#S3.F9)\), a larger magnitude in the first derivative term \(i\.e\., a steeper profile\) led to a larger contribution to the corresponding𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}component\.[24](https://arxiv.org/html/2605.08109#bib.bib13)The remaining features had less pronounced effects on model outputs and behaved in less predictable ways\. It is possible that our current parametrization does not fully represent all variance in the lift force\. Additional features such as hydraulic diameter or a modified particle blockage ratio defined relative to cross\-sectional area rather than a single characteristic dimension may improve model performance, and both can be defined even for complex geometries\.

![Two vertically stacked SHAP summary plots showing feature impact on model output for x and y lift force components. Features are listed on the y-axis, SHAP values on the x-axis, and color represents feature value from low (blue) to high (pink). In the top plot, w_x is the most influential feature, where high values (pink) correlate with positive SHAP values and low values (blue) with negative ones. In the bottom plot, w_y is most influential with a similar trend. In both plots, the Reynolds number (Re_p) and higher-order velocity derivatives are clustered tightly around the zero line, indicating they have significantly less impact on the model predictions than the primary velocity components.](https://arxiv.org/html/2605.08109v1/shap-2.png)Figure 9:Shapley values forCLxC\_\{\\texttt\{L\}\_\{x\}\}\(top\) andCLyC\_\{\\texttt\{L\}\_\{y\}\}\(bottom\), calculated using theSHAPlibrary for python on 500 samples from the rectangular test dataℛt​s\\mathcal\{R\}^\{ts\}\.

## 4Discussion

### 4\.1Model performance

Even when trained on rectangular data alone, our model was still able to provide reasonable predictions of𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}vectors in the other geometries\. This is qualitatively confirmed in[Fig\.˜7](https://arxiv.org/html/2605.08109#S3.F7), where the model was only trained on rectangular geometries but successfully learned much of the behaviour of𝐂Lx\\mathbf\{C\}\_\{\\texttt\{L\}\_\{x\}\}and𝐂Ly\\mathbf\{C\}\_\{\\texttt\{L\}\_\{y\}\}within the other channel geometries\. These results, together with the downstream simulation results in[Section˜2\.4](https://arxiv.org/html/2605.08109#S2.SS4), lead us to conclude that our parametrization can successfully predict lift force without requiring geometric parameters, and that this results in models which can generalize across geometries\.

An obvious caveat to the above claim is that it appears adding new geometries to the training data decreased performance on the existing geometries \([Fig\.˜6](https://arxiv.org/html/2605.08109#S3.F6)\)\. This suggests distributional shifts between the datasets\. Adversarial validation experiments with and without target values indicated that this wasdatadrift and notconceptdrift \([Fig\.˜S3](https://arxiv.org/html/2605.08109#Ax1.F3)\)\. In other words, the feature\-target relationship \([Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8)\) stayed consistent between datasets, but the distributions of the features themselves changed\. The data drift was particularly striking in the triangular and trapezoidal datasets, which were also the hardest geometries for the rectangular\-only model to predict \([Fig\.˜6](https://arxiv.org/html/2605.08109#S3.F6)\)\. It is difficult to determine if this input data drift is an artifact of simulation differences or a result of some unintended geometry dependence in our formulation \([Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8)\)\. For the trapezoidal data it seems likely that the distributional shift was a case of the former, as the dataset contained particle sizes, midline velocities, and channel heights that the other three datasets did not have\.

### 4\.2Simulation in curved channels

Our model successfully predicted particle migration even in channels with curves, corners, and non\-constant cross\-sections\. In such channels, particle migration is understood to depend partly on the development of secondary flows and the resulting Dean force\.[28](https://arxiv.org/html/2605.08109#bib.bib38)These forces were not present in the training data, which only contained data from straight channels\. Additionally, our model was only trained to predict lift forces in cross\-sectional directions \(theX​YXY\-plane, in a straight channel\), but particles in non\-straight channels experience lift forces outside of the cross\-sectional plane\. It is noteworthy that, given these limitations, the combination of our lift force model and rotational mapping technique proved robust enough to accurately predict particle trajectories, even in the presence of the complex fluid dynamics found in non\-straight channels\.

### 4\.3Error profile

Plotting the predicted values ofCLxC\_\{\\texttt\{L\}\_\{x\}\}andCLyC\_\{\\texttt\{L\}\_\{y\}\}against their actual values shows that the relative errors increase as the components approach zero \([Fig\.˜10\(a\)](https://arxiv.org/html/2605.08109#S4.F10.sf1)\)\. In other words, the smaller the𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}component, the harder it was for the model to predict\. This led to a spatial correlation in the model errors: the largest errors occurred either close to the channel centre or at a specific intermediate distance between the centre and the walls \([Fig\.˜10\(b\)](https://arxiv.org/html/2605.08109#S4.F10.sf2)\)\. These are both regions in which lift forces become vanishingly small, making them more difficult to predict\. At the channel centre, vanishingly small forces are from the symmetry of the velocity profile\. At the intermediate distance, they are due to a balance between shear gradient lift forces and wall\-induced lift and it is at these lateral equilibrium positions where particles ultimately focus during inertial migration\.[8](https://arxiv.org/html/2605.08109#bib.bib1),[46](https://arxiv.org/html/2605.08109#bib.bib7)\. The downstream simulations \([Fig\.˜8](https://arxiv.org/html/2605.08109#S3.F8)\) suggested that, practically, these errors were not large enough to prevent particles from stabilizing at their eventual equilibrium positions within the simulated IMDs\.

\(a\)![Two log-log scatter plots comparing predicted lift force coefficients to actual values for x and y components. Both plots show a dense cloud of blue data points clustered around a diagonal 1-to-1 dotted line ranging from 10^-9 to 10^-1. Prediction accuracy is highest at large magnitudes, with increased dispersion occurring for values below 10^-5.](https://arxiv.org/html/2605.08109v1/predictions.png)
\(b\)![Two vertically stacked plots showing model error versus relative distance from the channel center. The top plot (blue) shows angular error in degrees, which remains near zero across most distances but exhibits high spikes and dispersion at the center (0) and mid-wall regions (0.5 to 0.6). The bottom plot (pink) shows relative magnitude error on a log scale, and exhibits a similar profile.](https://arxiv.org/html/2605.08109v1/error-dist.png)

Figure 10:\(a\) Predicted lift coefficient vector components plotted against actual lift coefficient components\. \(b\) Distribution of angle and relative magnitude errors within a square\\qty50\\micro×\\times\\qty50\\micromicrochannel\. The relative distance from the channel centre is the relative distance of a particle along a line from the channel centre, through that particle, to the channel wall\. This distance is zero at the centre of the channel and unity at any wall\.To avoid explicit dependence on geometric parameters, the non\-dimensionalization step used in[Eq\.˜7](https://arxiv.org/html/2605.08109#S2.E7)differed slightly from literature and as a consequence ourCLxC\_\{\\texttt\{L\}\_\{x\}\}andCLyC\_\{\\texttt\{L\}\_\{y\}\}values spanned more orders of magnitude \(from1×10−101\\text\{\\times\}\{10\}^\{\-10\}to11\)\. Designing a loss function that is appropriate for a large range of extremely small target values centred around zero is challenging, as both absolute and relative errors have large drawbacks in such a setting\.[5](https://arxiv.org/html/2605.08109#bib.bib11)To reduce the error on small𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}we tried several approaches:\(a\)using relative loss functions, which led to gradient explosions and non\-convergence;\(b\)normalizing the target data, which lowered the upper limit ofCLxC\_\{\\texttt\{L\}\_\{x\}\}andCLyC\_\{\\texttt\{L\}\_\{y\}\}values but did not affect the majority of the data close to zero;\(c\)transforming the target data into logarithmic space, which introduced some challenges with negative and zero values, and resulted in worse models; and\(d\)predicting angle and magnitude instead of vector components, which improvedϕ50e​r​r\\phi^\{err\}\_\{50\}and∥𝐂L∥50e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}scores, but resulted in many more outliers and prediction artifacts, and thus much higherϕ90e​r​r\\phi^\{err\}\_\{90\}and∥𝐂L∥90e​r​r\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{90\}scores\.

## 5Conclusions

IMDs offer low\-cost, high\-throughput alternative techniques for many traditional particle\- and cell\-separation tasks, but simulating them requires being able to predict particle migration, and thus particle lift forces, under a variety of geometric conditions\. We exploited the underlying scaling laws governing particle migration to develop a novel model parametrization that has no explicit dependence on channel geometries\. We demonstrated how FCNN models using our parametrization can generalize to geometries that were not seen in the training data\. We further showed that these models can be readily incorporated into particle tracing simulations in software such asCOMSOL, and result in particle trajectories which match physical experiments from literature\. It is our hope that the model trained herein can be immediately useful to researchers running simulations, and that this alternative formulation for the lift force will open the door to further novel approaches as the AI and deep learning fields advance\.

## Author Contributions

Jesse Ward\-Bond:Conceptualization, Methodology, Investigation, Software, Visualization, Writing \- Original Draft Creation\.Ali Mashhadian:Conceptualization, Methodology, Data Curation, Simulation, Validation\.Timothy Chan:Conceptualization, Supervision, Writing \- Review & Editing\.Edmond Young:Conceptualization, Supervision, Writing \- Review & Editing\.

## Conflicts of interest

There are no conflicts to declare\.

## Data availability

## Acknowledgements

This research was supported by the NSERC Discovery Grant \(RGPIN\-2019\-05885\) held by Edmond Young and by the Ontario Graduate Scholarship, Queen Elizabeth II Graduate Scholarship in Science & Technology, and Vector Scholarship in AI held by Jesse Ward\-Bond\.

## References

- M\. Akhbari and M\. Nasr Esfahany \(2024\)A correlation\-based approach for fast prediction of the equilibrium position of particles within a spiral microchannel\.Indian Chem\. Eng\.,pp\. 1–17\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p7.1),[§1](https://arxiv.org/html/2605.08109#S1.p7.1.2.1)\.
- E\. S\. Asmolov, A\. L\. Dubov, T\. V\. Nizkaya, J\. Harting, and O\. I\. Vinogradova \(2018\)Inertial focusing of finite\-size particles in microchannels\.J\. Fluid Mech\.840,pp\. 613–630\.External Links:[Document](https://dx.doi.org/10.1017/jfm.2018.95)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p5.1.2.1)\.
- J\. Bai, F\. Lu, K\. Zhang,et al\.\(2019\)ONNX: Open Neural Network Exchange\.GitHub\.Note:[https://github\.com/onnx/onnx](https://github.com/onnx/onnx)Cited by:[§2\.4](https://arxiv.org/html/2605.08109#S2.SS4.p1.1.1.1)\.
- S\. R\. Bazaz, A\. Mashhadian, A\. Ehsani, S\. C\. Saha, T\. Krüger, and M\. E\. Warkiani \(2020\)Computational inertial microfluidics: A review\.Lab Chip20\(6\),pp\. 1023–1048\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1),[§1](https://arxiv.org/html/2605.08109#S1.p5.1.1.1)\.
- A\. Botchkarev \(2019\)A New Typology Design of Performance Metrics to Measure Errors in Machine Learning Regression Algorithms\.Interdisciplinary Journal of Information, Knowledge, and Management14,pp\. 045–076\.External Links:[Document](https://dx.doi.org/10.28945/4184),[Link](https://doi.org/10.28945%2F4184)Cited by:[§4\.3](https://arxiv.org/html/2605.08109#S4.SS3.p2.5.1.1)\.
- E\. Buckingham \(1914\)On physically similar systems; illustrations of the use of dimensional equations\.Phys\. Rev\.4\(4\),pp\. 345\.Cited by:[§2\.1](https://arxiv.org/html/2605.08109#S2.SS1.p7.2.1.1)\.
- J\. S\. Dai \(2015\)Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections\.Mech\. Mach\. Theory92,pp\. 144–152\.Cited by:[§S3](https://arxiv.org/html/2605.08109#Ax1.SS3.p2.7.1.1),[§2\.4](https://arxiv.org/html/2605.08109#S2.SS4.p2.6.1.1)\.
- D\. Di Carlo, J\. F\. Edd, K\. J\. Humphry, H\. A\. Stone, and M\. Toner \(2009\)Particle segregation and dynamics in confined flows\.Phys\. Rev\. Lett\.102\(9\),pp\. 094503\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p6.1.2.1),[§4\.3](https://arxiv.org/html/2605.08109#S4.SS3.p1.3.1.1)\.
- D\. Di Carlo, D\. Irimia, R\. G\. Tompkins, and M\. Toner \(2007\)Continuous inertial focusing, ordering, and separation of particles in microchannels\.Proc\. Natl\. Acad\. Sci\. U\. S\. A\.104\(48\),pp\. 18892–18897\.External Links:[Document](https://dx.doi.org/10.1073/pnas.0704958104)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p4.1)\.
- V\. Godbole, G\. E\. Dahl, J\. Gilmer, C\. J\. Shallue, and Z\. Nado \(2023\)Deep Learning Tuning Playbook\.Note:Version 1\.0External Links:[Link](http://github.com/google-research/tuning_playbook)Cited by:[§2\.3](https://arxiv.org/html/2605.08109#S2.SS3.p1.9)\.
- A\. Grimes, D\. N\. Breslauer, M\. Long, J\. Pegan, L\. P\. Lee, and M\. Khine \(2008\)Shrinky\-Dink microfluidics: rapid generation of deep and rounded patterns\.Lab Chip8\(1\),pp\. 170–172\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p2.1.2.1)\.
- G\. Guan, L\. Wu, A\. A\. Bhagat, Z\. Li, P\. C\. Chen, S\. Chao, C\. J\. Ong, and J\. Han \(2013\)Spiral microchannel with rectangular and trapezoidal cross\-sections for size based particle separation\.Sci\. Rep\.3\(1\),pp\. 1475\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1)\.
- D\. J\. Guckenberger, T\. E\. De Groot, A\. M\. Wan, D\. J\. Beebe, and E\. W\. Young \(2015\)Micromilling: a method for ultra\-rapid prototyping of plastic microfluidic devices\.Lab Chip15\(11\),pp\. 2364–2378\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p2.1.1.1)\.
- E\. Hamdi, R\. Dezhkam, A\. Shamloo, and A\. Mashhadian \(2022\)MicroAI: a machine learning tool for fast calculation of lift coefficients in microchannels\.External Links:2210\.11591,[Link](https://arxiv.org/abs/2210.11591)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p7.1)\.
- K\. Hood, S\. Lee, and M\. Roper \(2015\)Inertial migration of a rigid sphere in three\-dimensional Poiseuille flow\.J\. Fluid Mech\.765,pp\. 452–479\.External Links:[Document](https://dx.doi.org/10.1017/jfm.2014.739)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p5.1.2.1)\.
- S\. C\. Hur, T\. Z\. Brinckerhoff, C\. M\. Walthers, J\. C\. Dunn, and D\. Di Carlo \(2012\)Label\-free enrichment of adrenal cortical progenitor cells using inertial microfluidics\.PLOS ONE,pp\. 46550\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1)\.
- J\. Hwang, Y\. H\. Cho, M\. S\. Park, and B\. H\. Kim \(2019\)Microchannel fabrication on glass materials for microfluidic devices\.International Journal of Precision Engineering and Manufacturing20\(3\),pp\. 479–495\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p2.1.1.1)\.
- H\. Jeon, T\. Kwon, J\. Yoon, and J\. Han \(2022\)Engineering a deformation\-free plastic spiral inertial microfluidic system for CHO cell clarification in biomanufacturing\.Lab Chip22\(2\),pp\. 272–285\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.3.1)\.
- F\. Jiang and N\. Xiang \(2022\)Integrated microfluidic handheld cell sorter for high\-throughput label\-free malignant tumor cell sorting\.Anal\. Chem\.94\(3\),pp\. 1859–1866\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.3.1)\.
- U\. Kim, J\. Kwon, T\. Kim, and Y\. Cho \(2022\)Particle focusing in a straight microchannel with non\-rectangular cross\-section\.Micromachines13\(2\),pp\. 151\.Cited by:[Figure S6](https://arxiv.org/html/2605.08109#Ax1.F6)\.
- M\. G\. Lee, S\. Choi, and J\. Park \(2011\)Inertial separation in a contraction–expansion array microchannel\.J\. Chromatogr\. A1218\(27\),pp\. 4138–4143\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1),[item \(b\)](https://arxiv.org/html/2605.08109#S2.I1.i2.1.1.1)\.
- G\. Li, G\. H\. McKinley, and A\. M\. Ardekani \(2015\)Dynamics of particle migration in channel flow of viscoelastic fluids\.J\. Fluid Mech\.785,pp\. 486–505\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p6.1.2.1)\.
- C\. Liu, G\. Hu, X\. Jiang, and J\. Sun \(2015\)Inertial focusing of spherical particles in rectangular microchannels over a wide range of Reynolds numbers\.Lab Chip15\(4\),pp\. 1168–1177\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p6.1.2.1),[§1](https://arxiv.org/html/2605.08109#S1.p6.1.3.1),[§2\.1](https://arxiv.org/html/2605.08109#S2.SS1.p2.6.1.1)\.
- S\. M\. Lundberg and S\. Lee \(2017\)A unified approach to interpreting model predictions\.InProceedings of the 31st International Conference on Neural Information Processing Systems,NIPS’17,Red Hook, NY, USA,pp\. 4768–4777\.External Links:ISBN 9781510860964Cited by:[§2\.3](https://arxiv.org/html/2605.08109#S2.SS3.p4.1.3.1),[§3\.3](https://arxiv.org/html/2605.08109#S3.SS3.p1.5.1.1)\.
- D\. McIntyre, A\. Lashkaripour, P\. Fordyce, and D\. Densmore \(2022\)Machine learning for microfluidic design and control\.Lab Chip22\(16\),pp\. 2925–2937\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p7.1.1.1)\.
- R\. Moloudi, S\. Oh, C\. Yang, M\. Ebrahimi Warkiani, and M\. W\. Naing \(2018\)Inertial particle focusing dynamics in a trapezoidal straight microchannel: Application to particle filtration\.Microfluid\. Nanofluid\.22,pp\. 1–14\.Cited by:[Figure S5](https://arxiv.org/html/2605.08109#Ax1.F5)\.
- Y\. Nesterov \(1983\)A method of solving a convex programming problem with convergence rateOO\(1/k2\)\.Proc\. USSR Acad\. Sci\.269,pp\. 3\.Cited by:[§2\.3](https://arxiv.org/html/2605.08109#S2.SS3.p1.2.1.1)\.
- N\. Nivedita, P\. Ligrani, and I\. Papautsky \(2017\)Dean flow dynamics in low\-aspect ratio spiral microchannels\.Sci\. Rep\.7\(1\),pp\. 44072\.Cited by:[§4\.2](https://arxiv.org/html/2605.08109#S4.SS2.p1.1.1.1)\.
- B\. Owen \(2024\)Accelerating the development of inertial microfluidic devices using numerical modelling and machine learning\.Front\. Lab Chip Technol\.3,pp\. 1328004\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p7.1.1.1)\.
- J\. Park, S\. Song, and H\. Jung \(2009\)Continuous focusing of microparticles using inertial lift force and vorticity via multi\-orifice microfluidic channels\.Lab Chip9\(7\),pp\. 939–948\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1)\.
- M\. Safari, P\. Abbasi, S\. Momeni, M\. S\. Farahani, and H\. Safari \(2024\)SpiralDesigner: An AI\-assisted design interface for efficient separation of neutrally buoyant and non\-buoyant particles using spiral microfluidic devices\.Chem\. Eng\. Sci\.297,pp\. 120301\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p7.1),[§1](https://arxiv.org/html/2605.08109#S1.p7.1.2.1)\.
- G\. Segré and A\. Silberberg \(1961\)Radial particle displacements in Poiseuille flow of suspensions\.Nature189\(4760\),pp\. 209–210\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p4.1)\.
- A\. Shamloo and A\. Mashhadian \(2018\)Inertial particle focusing in serpentine channels on a centrifugal platform\.Phys\. Fluids30\(1\),pp\. 012002\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p6.1.2.1),[§1](https://arxiv.org/html/2605.08109#S1.p6.1.3.1),[§2\.2](https://arxiv.org/html/2605.08109#S2.SS2.p1.8.2.1.1)\.
- S\. K\. Sia and G\. M\. Whitesides \(2003\)Microfluidic devices fabricated in poly \(dimethylsiloxane\) for biological studies\.Electrophoresis24\(21\),pp\. 3563–3576\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p2.1.1.1)\.
- J\. Su, X\. Chen, Y\. Zhu, and G\. Hu \(2021\)Machine learning assisted fast prediction of inertial lift in microchannels\.Lab Chip21\(13\),pp\. 2544–2556\.Cited by:[§S2](https://arxiv.org/html/2605.08109#Ax1.SS2.p1.1),[§1](https://arxiv.org/html/2605.08109#S1.p7.1),[Figure 3](https://arxiv.org/html/2605.08109#S2.F3.12.12.11.11.6.1.1),[Figure 3](https://arxiv.org/html/2605.08109#S2.F3.6.6.5.5.6.1.1),[Figure 3](https://arxiv.org/html/2605.08109#S2.F3.9.9.8.8.6.1.1),[§2\.1](https://arxiv.org/html/2605.08109#S2.SS1.p2.6.1.1),[§2\.2](https://arxiv.org/html/2605.08109#S2.SS2.p1.8.1.1),[§2\.3](https://arxiv.org/html/2605.08109#S2.SS3.p3.3),[Figure 5](https://arxiv.org/html/2605.08109#S3.F5),[§3\.1](https://arxiv.org/html/2605.08109#S3.SS1.p1.1)\.
- J\. Su, X\. Zheng, and G\. Hu \(2023\)New explicit formula for inertial lift in confined flows\.Phys\. Fluids35\(9\),pp\. 092010\.External Links:[Document](https://dx.doi.org/10.1063/5.0168147)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p5.1.2.1)\.
- I\. Sutskever, J\. Martens, G\. Dahl, and G\. Hinton \(2013\)On the importance of initialization and momentum in deep learning\.InProceedings of the 30th International Conference on Machine Learning,S\. Dasgupta and D\. McAllester \(Eds\.\),Proceedings of Machine Learning Research, Vol\.28,Atlanta, Georgia, USA,pp\. 1139–1147\.External Links:[Link](https://proceedings.mlr.press/v28/sutskever13.html)Cited by:[§2\.3](https://arxiv.org/html/2605.08109#S2.SS3.p1.2.1.1)\.
- W\. Tang, S\. Zhu, D\. Jiang, L\. Zhu, J\. Yang, and N\. Xiang \(2020\)Channel innovations for inertial microfluidics\.Lab Chip20\(19\),pp\. 3485–3502\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p3.1.1.1)\.
- A\. Volpe, P\. Paiè, A\. Ancona, and R\. Osellame \(2019\)Polymeric fully inertial lab\-on\-a\-chip with enhanced\-throughput sorting capabilities\.Microfluid\. Nanofluid\.23\(3\),pp\. 37\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.3.1)\.
- M\. E\. Warkiani, B\. L\. Khoo, L\. Wu, A\. K\. P\. Tay, A\. A\. S\. Bhagat, J\. Han, and C\. T\. Lim \(2016\)Ultra\-fast, label\-free isolation of circulating tumor cells from blood using spiral microfluidics\.Nat\. Protoc\.11\(1\),pp\. 134–148\.Cited by:[Figure S8](https://arxiv.org/html/2605.08109#Ax1.F8)\.
- Z\. Wu, Y\. Chen, M\. Wang, and A\. J\. Chung \(2016\)Continuous inertial microparticle and blood cell separation in straight channels with local microstructures\.Lab Chip16\(3\),pp\. 532–542\.Cited by:[Figure 8](https://arxiv.org/html/2605.08109#S3.F8),[§3\.2](https://arxiv.org/html/2605.08109#S3.SS2.p2.1.5.1)\.
- N\. Xiang and Z\. Ni \(2022\)Inertial microfluidics: current status, challenges, and future opportunities\.Lab Chip22\(24\),pp\. 4792–4804\.External Links:[Document](https://dx.doi.org/10.1039/D2LC00722C)Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1)\.
- D\. H\. Yoon, J\. B\. Ha, Y\. K\. Bahk, T\. Arakawa, S\. Shoji, and J\. S\. Go \(2009\)Size\-selective separation of micro beads by utilizing secondary flow in a curved rectangular microchannel\.Lab Chip9\(1\),pp\. 87–90\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1)\.
- J\. Zhang, S\. Yan, R\. Sluyter, W\. Li, G\. Alici, and N\. Nguyen \(2014\)Inertial particle separation by differential equilibrium positions in a symmetrical serpentine micro\-channel\.Sci\. Rep\.4\(1\),pp\. 4527\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1),[item \(b\)](https://arxiv.org/html/2605.08109#S2.I1.i2.1.2.1),[Figure 8](https://arxiv.org/html/2605.08109#S3.F8),[§3\.2](https://arxiv.org/html/2605.08109#S3.SS2.p2.1.10.1)\.
- J\. Zhang, S\. Yan, D\. Yuan, G\. Alici, N\. Nguyen, M\. E\. Warkiani, and W\. Li \(2016\)Fundamentals and applications of inertial microfluidics: a review\.Lab Chip16\(1\),pp\. 10–34\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.1.1)\.
- J\. Zhou and I\. Papautsky \(2013\)Fundamentals of inertial focusing in microchannels\.Lab Chip13\(6\),pp\. 1121–1132\.Cited by:[§1](https://arxiv.org/html/2605.08109#S1.p1.1.2.1),[§4\.3](https://arxiv.org/html/2605.08109#S4.SS3.p1.3.1.1)\.
- Y\. Zhou, Z\. Ma, and Y\. Ai \(2018\)Sheathless inertial cell focusing and sorting with serial reverse wavy channel structures\.Microsyst\. Nanoeng\.4\(1\),pp\. 5\.Cited by:[Figure S7](https://arxiv.org/html/2605.08109#Ax1.F7),[§2\.4](https://arxiv.org/html/2605.08109#S2.SS4.p1.1)\.

## Supplementary Information

### S1Taylor Series Approximation

[Figure˜S1](https://arxiv.org/html/2605.08109#Ax1.F1)shows that a second\-order Taylor series expansion can almost completely recover the velocity profile around the circumference of an “imaginary” \(non\-interacting\) spherical particle within a straight rectangular channel\.

![Refer to caption](https://arxiv.org/html/2605.08109v1/taylor.png)Figure S1:The second\-order Taylor series expansion of the local velocity profile around the circumference of a particle in a rectangular channel\.
### S2𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}Conversion

To convert from the𝐂L\\mathbf\{C\}\_\{\\texttt\{L\}\}formulation used inSuet al\.[35](https://arxiv.org/html/2605.08109#bib.bib5)to our current methodology, we use

𝐂L=𝐂LS​u⋅a2​Um2w2​H2,\\mathbf\{C\}\_\{\\texttt\{L\}\}=\\mathbf\{C\}\_\{\\texttt\{L\}\}^\{Su\}\\cdot\\frac\{a^\{2\}U\_\{m\}^\{2\}\}\{w^\{2\}H^\{2\}\},\(S1\)
whereUmU\_\{m\}is the maximum \(centerline\) velocity in the channel\.

### S3Generalization to Arbitrary 3D Orientations

Note that in the following, we use\(x,y,z\)\(x,y,z\)to represent theglobal\(lab referenced\) coordinate system\.

As established in[Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8), the model formulation assumes that velocity vectors in the undisturbed flow are strictly aligned with thezz\-direction, rendering allzz\-gradients zero\. While this performs well for fully developed flows in straight channels, it is insufficient for particle tracing in channels with arbitrary orientations \(e\.g\., curved or serpentine channels\) where the local fluid velocity vector continuously changes direction\. To address this, we introduce a rotational mapping process to generalize the model for any 3D orientation using Rodrigues’ rotation formula\.[7](https://arxiv.org/html/2605.08109#bib.bib33)\. This mapping aligns the local fluid velocity vector𝐮=\{u,v,w\}⊤\\mathbf\{u\}=\\\{u,v,w\\\}^\{\\top\}with the globalzz\-direction \(represented by the unit vector𝐧^z=\{0,0,1\}⊤\\mathbf\{\\hat\{n\}\}\_\{z\}=\\\{0,0,1\\\}^\{\\top\}\)\. Once rotated, we can calculate the𝐂LX​Y′\\mathbf\{C\}^\{\\prime\}\_\{\\texttt\{L\}\_\{XY\}\}—the lift force coefficient in the rotated reference frame—and then perform the inverse rotation to map this back to𝐂LX​Y​Z\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XYZ\}\}, the 3D lift force vector in the original reference frame\.

We begin by constructing the rotation matrix,𝐑M\\mathbf\{R\}\_\{M\}\. The normalized velocity vector is defined as

𝐧^=𝐮‖𝐮‖=\(u‖𝐮‖,v‖𝐮‖,w‖𝐮‖\)\.\\mathbf\{\\hat\{n\}\}=\\frac\{\\mathbf\{u\}\}\{\\\|\\mathbf\{u\}\\\|\}=\\left\(\\frac\{u\}\{\\\|\\mathbf\{u\}\\\|\},\\frac\{v\}\{\\\|\\mathbf\{u\}\\\|\},\\frac\{w\}\{\\\|\\mathbf\{u\}\\\|\}\\right\)\.\(S2\)The axis of rotation is therefore𝐧^×𝐧^z\\mathbf\{\\hat\{n\}\}\\times\\mathbf\{\\hat\{n\}\}\_\{z\}, and the angle of rotation is𝐧^⋅𝐧^z\\mathbf\{\\hat\{n\}\}\\cdot\\mathbf\{\\hat\{n\}\}\_\{z\}\. We then formulate the skew\-symmetric cross\-product matrix𝐒C\\mathbf\{S\}\_\{C\}as:

𝐒C=\[00−u‖𝐮‖00−v‖𝐮‖u‖𝐮‖v‖𝐮‖0\]\\mathbf\{S\}\_\{C\}=\\begin\{bmatrix\}0&0&\-\\frac\{u\}\{\\\|\\mathbf\{u\}\\\|\}\\\\ 0&0&\-\\frac\{v\}\{\\\|\\mathbf\{u\}\\\|\}\\\\ \\frac\{u\}\{\\\|\\mathbf\{u\}\\\|\}&\\frac\{v\}\{\\\|\\mathbf\{u\}\\\|\}&0\\end\{bmatrix\}\(S3\)
The rotation matrix𝐑M\\mathbf\{R\}\_\{M\}is then defined as:

𝐑M=𝐈\+𝐒C\+\(1−𝐧^⋅𝐧^z‖𝐧^×𝐧^z‖2\)​𝐒C2=𝐈\+𝐒C\+\(1−w‖𝐮‖\(u‖𝐮‖\)2\+\(v‖𝐮‖\)2\)​𝐒C2\\begin\{split\}\\mathbf\{R\}\_\{M\}&=\\mathbf\{I\}\+\\mathbf\{S\}\_\{C\}\+\\left\(\\frac\{1\-\\mathbf\{\\hat\{n\}\}\\cdot\\mathbf\{\\hat\{n\}\}\_\{z\}\}\{\\\|\\mathbf\{\\hat\{n\}\}\\times\\mathbf\{\\hat\{n\}\}\_\{z\}\\\|^\{2\}\}\\right\)\\mathbf\{S\}\_\{C\}^\{2\}\\\\ &=\\mathbf\{I\}\+\\mathbf\{S\}\_\{C\}\+\\left\(\\frac\{1\-\\frac\{w\}\{\\\|\\mathbf\{u\}\\\|\}\}\{\\left\(\\frac\{u\}\{\\\|\\mathbf\{u\}\\\|\}\\right\)^\{2\}\+\\left\(\\frac\{v\}\{\\\|\\mathbf\{u\}\\\|\}\\right\)^\{2\}\}\\right\)\\mathbf\{S\}\_\{C\}^\{2\}\\end\{split\}\(S4\)
where𝐈\\mathbf\{I\}is the3×33\\times 3identity matrix\.

LetU=‖𝐮‖U=\\\|\\mathbf\{u\}\\\|denote the magnitude of the velocity vector\. The first\-order gradient vector𝐆1\\mathbf\{G\}\_\{1\}and second\-order gradient tensor𝐆2\\mathbf\{G\}\_\{2\}of the velocity field \(in the global coordinate system\) are defined as:

𝐆1=\[UxUyUz\],𝐆2=\[Ux​xUx​yUx​zUy​xUy​yUy​zUz​xUz​yUz​z\]\\mathbf\{G\}\_\{1\}=\\begin\{bmatrix\}U\_\{x\}\\\\ U\_\{y\}\\\\ U\_\{z\}\\end\{bmatrix\},\\quad\\mathbf\{G\}\_\{2\}=\\begin\{bmatrix\}U\_\{xx\}&U\_\{xy\}&U\_\{xz\}\\\\ U\_\{yx\}&U\_\{yy\}&U\_\{yz\}\\\\ U\_\{zx\}&U\_\{zy\}&U\_\{zz\}\\end\{bmatrix\}\(S5\)
Applying𝐑M\\mathbf\{R\}\_\{M\}, we map these gradients into thex​yxy\-plane\. We use𝐮′=\{u′,v′,w′\}\\mathbf\{u\}^\{\\prime\}=\\left\\\{u^\{\\prime\},v^\{\\prime\},w^\{\\prime\}\\right\\\}to represent the rotated velocity vector\. This vector is in the direction of𝐧^𝐳\\mathbf\{\\hat\{n\}\_\{z\}\}, and thus𝐮′=\{0,0,w′\}\\mathbf\{u\}^\{\\prime\}=\\left\\\{0,0,w^\{\\prime\}\\right\\\}\. The mapped gradients are:

𝐆1′=𝐑M​𝐆1=\[wx′wy′0\]\\displaystyle\\mathbf\{G\}^\{\\prime\}\_\{1\}=\\mathbf\{R\}\_\{M\}\\mathbf\{G\}\_\{1\}=\\begin\{bmatrix\}w^\{\\prime\}\_\{x\}\\\\ w^\{\\prime\}\_\{y\}\\\\ 0\\end\{bmatrix\}\(S6\)𝐆2′=𝐑M​𝐆2​𝐑M⊤=\[wx​x′wx​y′0wy​x′wy​y′0000\]\\displaystyle\\mathbf\{G\}^\{\\prime\}\_\{2\}=\\mathbf\{R\}\_\{M\}\\mathbf\{G\}\_\{2\}\\mathbf\{R\}\_\{M\}^\{\\top\}=\\begin\{bmatrix\}w^\{\\prime\}\_\{xx\}&w^\{\\prime\}\_\{xy\}&0\\\\ w^\{\\prime\}\_\{yx\}&w^\{\\prime\}\_\{yy\}&0\\\\ 0&0&0\\end\{bmatrix\}\(S7\)
Because the gradients of velocity along the streamline are negligible compared to the cross\-sectional gradients \(X,YX,Y\), all terms containingzzderivatives are neglected\. We can therefore reduce the mapped gradients to the 2D planar features required by the model:

𝐆1′=\[wx′wy′\],𝐆2′=\[wx​x′wx​y′wy​x′wy​y′\]\\mathbf\{G\}^\{\\prime\}\_\{1\}=\\begin\{bmatrix\}w^\{\\prime\}\_\{x\}\\\\ w^\{\\prime\}\_\{y\}\\end\{bmatrix\},\\quad\\mathbf\{G\}^\{\\prime\}\_\{2\}=\\begin\{bmatrix\}w^\{\\prime\}\_\{xx\}&w^\{\\prime\}\_\{xy\}\\\\ w^\{\\prime\}\_\{yx\}&w^\{\\prime\}\_\{yy\}\\end\{bmatrix\}\(S8\)
These spatial derivatives serve as inputs to the neural network\. The planar lift force coefficient is predicted using:

𝐂LX​Y′=f​\(R​ep′,wx′​aw′,wy′​aw′,wx​x′​a2w′,wy​y′​a2w′,wx​y′​a2w′\)\\mathbf\{C\}^\{\\prime\}\_\{\\texttt\{L\}\_\{XY\}\}=f\\left\(Re^\{\\prime\}\_\{p\},\\frac\{w^\{\\prime\}\_\{x\}a\}\{w^\{\\prime\}\},\\frac\{w^\{\\prime\}\_\{y\}a\}\{w^\{\\prime\}\},\\frac\{w^\{\\prime\}\_\{xx\}a^\{2\}\}\{w^\{\\prime\}\},\\frac\{w^\{\\prime\}\_\{yy\}a^\{2\}\}\{w^\{\\prime\}\},\\frac\{w^\{\\prime\}\_\{xy\}a^\{2\}\}\{w^\{\\prime\}\}\\right\)\(S9\)
whereR​ep=ρ​w′​aμRe\_\{p\}=\\frac\{\\rho w^\{\\prime\}a\}\{\\mu\}, Finally, the resultant 3D lift force coefficient is recovered by applying the inverse rotation matrix𝐑M⊤\\mathbf\{R\}\_\{M\}^\{\\top\}, and the 3D lift force is calculated as:

𝐂LX​Y​Z\\displaystyle\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XYZ\}\}=𝐑M⊤​𝐂LX​Y′\\displaystyle=\\mathbf\{R\}\_\{M\}^\{\\top\}\\mathbf\{C\}^\{\\prime\}\_\{\\texttt\{L\}\_\{XY\}\}\(S10\)𝐅LX​Y​Z\\displaystyle\\mathbf\{F\}\_\{\\texttt\{L\}\_\{XYZ\}\}=ρ​U2​a2​𝐂LX​Y​Z\\displaystyle=\\rho U^\{2\}a^\{2\}\\mathbf\{C\}\_\{\\texttt\{L\}\_\{XYZ\}\}\(S11\)

### S4Principal Component Analysis

Plotting the first two principal components of the training data shows that the triangular and trapezoidal datasets have wider distributions than the rectangular and semicircular datasets \([Fig\.˜S2](https://arxiv.org/html/2605.08109#Ax1.F2)\)\.

![Refer to caption](https://arxiv.org/html/2605.08109v1/pca_geometry_analysis.png)Figure S2:The first two principal components of the training data, separated by geometry type and coloured by prediction error\.
### S5Numerical generalization results

Table 1:Generalization of trained model to other geometries\. Values are reported as Median \(90th Percentile\)\. Models were retrained completely when additional geometries were added to the training set\.ℛ,𝒯,𝒮,𝒫\\mathcal\{R,T,S,P\}, represent the rectangular, triangular, semicircular, and trapezoidal datasets, respectively\. Superscriptst​rtrandt​stsrepresent training and testing subsets\.MetricTraining SetTest Setℛt​s\\mathcal\{R\}^\{ts\}𝒯t​s\\mathcal\{T\}^\{ts\}𝒮t​s\\mathcal\{S\}^\{ts\}𝒫t​s\\mathcal\{P\}^\{ts\}MSE\{ℛ\}t​r\\\{\\mathcal\{R\}\\\}^\{tr\}2\.27×10−62\.27\\text\{\\times\}\{10\}^\{\-6\}8\.83×10−58\.83\\text\{\\times\}\{10\}^\{\-5\}5\.18×10−65\.18\\text\{\\times\}\{10\}^\{\-6\}7\.44×10−57\.44\\text\{\\times\}\{10\}^\{\-5\}\{ℛ,𝒯\}t​r\\\{\\mathcal\{R,T\}\\\}^\{tr\}2\.38×10−62\.38\\text\{\\times\}\{10\}^\{\-6\}1\.53×10−61\.53\\text\{\\times\}\{10\}^\{\-6\}2\.85×10−62\.85\\text\{\\times\}\{10\}^\{\-6\}8\.22×10−58\.22\\text\{\\times\}\{10\}^\{\-5\}\{ℛ,𝒯,𝒮\}t​r\\\{\\mathcal\{R,T,S\}\\\}^\{tr\}2\.57×10−62\.57\\text\{\\times\}\{10\}^\{\-6\}1\.75×10−61\.75\\text\{\\times\}\{10\}^\{\-6\}2\.02×10−62\.02\\text\{\\times\}\{10\}^\{\-6\}1\.03×10−41\.03\\text\{\\times\}\{10\}^\{\-4\}\{ℛ,𝒯,𝒮,𝒫\}t​r\\\{\\mathcal\{R,T,S,P\}\\\}^\{tr\}3\.58×10−63\.58\\text\{\\times\}\{10\}^\{\-6\}3\.94×10−63\.94\\text\{\\times\}\{10\}^\{\-6\}2\.68×10−62\.68\\text\{\\times\}\{10\}^\{\-6\}3\.89×10−53\.89\\text\{\\times\}\{10\}^\{\-5\}ϕ50e​r​r​\(ϕ90e​r​r\)\\phi^\{err\}\_\{50\}\\ \(\\phi^\{err\}\_\{90\}\)\(\\unit\)\{ℛ\}t​r\\\{\\mathcal\{R\}\\\}^\{tr\}4\.6 \(20\.7\)7\.6 \(51\.0\)12\.8 \(47\.1\)5\.9 \(58\.8\)\{ℛ,𝒯\}t​r\\\{\\mathcal\{R,T\}\\\}^\{tr\}4\.7 \(22\.1\)3\.2 \(13\.5\)12\.3 \(47\.4\)6\.2 \(56\.7\)\{ℛ,𝒯,𝒮\}t​r\\\{\\mathcal\{R,T,S\}\\\}^\{tr\}5\.4 \(22\.3\)3\.3 \(14\.4\)9\.7 \(39\.3\)6\.9 \(50\.8\)\{ℛ,𝒯,𝒮,𝒫\}t​r\\\{\\mathcal\{R,T,S,P\}\\\}^\{tr\}7\.0 \(29\.1\)4\.8 \(20\.3\)11\.6 \(44\.8\)6\.2 \(46\.5\)∥𝐂L∥50e​r​r​\(∥𝐂L∥90e​r​r\)\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{50\}\\ \(\\lVert\\mathbf\{C\}\_\{\\texttt\{L\}\}\\rVert^\{err\}\_\{90\}\)\(%\)\{ℛ\}t​r\\\{\\mathcal\{R\}\\\}^\{tr\}8\.7 \(38\.3\)21\.2 \(106\.0\)20\.3 \(65\.6\)13\.0 \(85\.4\)\{ℛ,𝒯\}t​r\\\{\\mathcal\{R,T\}\\\}^\{tr\}9\.5 \(39\.9\)6\.6 \(24\.8\)19\.0 \(62\.8\)13\.7 \(81\.1\)\{ℛ,𝒯,𝒮\}t​r\\\{\\mathcal\{R,T,S\}\\\}^\{tr\}10\.4 \(51\.5\)7\.0 \(27\.4\)16\.2 \(53\.8\)15\.9 \(103\.3\)\{ℛ,𝒯,𝒮,𝒫\}t​r\\\{\\mathcal\{R,T,S,P\}\\\}^\{tr\}14\.9 \(71\.2\)11\.0 \(42\.3\)18\.1 \(63\.3\)12\.4 \(81\.4\)
### S6Adversarial validation

[Figure˜S3](https://arxiv.org/html/2605.08109#Ax1.F3)shows the results of adversarial validation between all pairs of datasets\. For each pair of datasets, we combined the two datasets and trained a histogram\-based gradient boosting classification tree to predict which dataset each point belonged to using only the input features from[Eq\.˜8](https://arxiv.org/html/2605.08109#S2.E8)\. An area under the curve \(AUC\) score of0\.50\.5indicates that the model is unable to distinguish between the two datasets\. Similarly, an AUC score of1\.01\.0shows that the model can perfectly separate the two datasets\. We performed adversarial validation between theaugmented\(rotated and flipped\) datasets\.

![Refer to caption](https://arxiv.org/html/2605.08109v1/adversarial.png)Figure S3:Area under the curve \(AUC\) scores during adversarial validation of all input datasets using a histogram\-based gradient boosting classification tree\.
### S7Additional Simulation Results

![Refer to caption](https://arxiv.org/html/2605.08109v1/s.png)Figure S4:Inertial particle focusing in a straight circular microchannel using our geometry\-free lift force model\. The model was trained on straight channels with rectangular, triangular, semicircular, and trapezoidal cross\-sections\. The average fluid velocity in the channel is\\qty0\.2\\per, the particle size is\\qty10\\micro, and the cross\-section diameter is\\qty100\\micro\. a\) Three\-dimensional view of particles movement in the channel\. b\) Vector field plot of the predicted inertial lift forces\. c\) Particle positions at the channel inlet\. d\) Particle positions at the channel outlet\.![Refer to caption](https://arxiv.org/html/2605.08109v1/p.png)Figure S5:Inertial particle focusing in a straight trapezoidal microchannel using our geometry\-free lift force model\. The model was trained on straight channels with rectangular, triangular, semicircular, and trapezoidal cross\-sections\. The flow rate in the channel is\\qty0\.2\\milli\\per, the particle size is\\qty20\\micro, and base, short height and long height of the trapezoid are\\qty200\\micro,\\qty50\\micro, and\\qty140\\micro, respectively\. b\) Experimental results from literature for a straight trapezoidal channel under similar conditions\. The green fluorescent lines represent the trajectories of the particles, and the white dashed lines delineate the channel walls\. “S” and “L” indicate the short and long walls of the trapezoidal cross section, respectively\. Adapted fromMoloudiet al\.[26](https://arxiv.org/html/2605.08109#bib.bib8)with permission from Springer Nature\. c\) Vector field plot of the predicted inertial lift forces\. d\) Particle positions at the channel inlet\. e\) Particle positions at the channel outlet\.![Refer to caption](https://arxiv.org/html/2605.08109v1/rh.png)Figure S6:Inertial particle focusing in a straight rhombus microchannel using our geometry\-free lift force model\. The model was trained on straight channels with rectangular, triangular, semicircular, and trapezoidal cross\-sections\. The flow rate in the channel is\\qty0\.1\\milli\\per, and the particle size is\\qty13\\micro\. The diagonals of the rhombus measure\\qty68\.4\\microand\\qty146\.2\\micro, respectively\. b\) Experimental results from literature for a straight rhombus channel under similar conditions\. Adapted from "Particle Focusing in a Straight Microchannel with Non\-Rectangular Cross\-Section" byKimet al\.[20](https://arxiv.org/html/2605.08109#bib.bib35)under CC BY 4\.0 \([https://creativecommons\.org/licenses/by/4\.0/](https://creativecommons.org/licenses/by/4.0/)\)\. Image is a cropped version of the original\. c\) Vector field plot of the predicted inertial lift forces\. d\) Particle positions at the channel inlet\. e\) Particle positions at the channel outlet\.![Refer to caption](https://arxiv.org/html/2605.08109v1/reverse_wavy.png)Figure S7:Inertial separation of\\qty3\\micro\(blue\),\\qty10\\micro\(green\), and\\qty15\\micro\(red\) particles in “reverse wavy” channels\. Flow rates were\\qty197\.60\\micro\\per\. a\) Experimental results from literature for a reverse wavy channel under similar conditions\. Adapted from "A high\-throughput micropump with a wide range of flow rates" byZhouet al\.[47](https://arxiv.org/html/2605.08109#bib.bib32)under CC BY 4\.0 \([https://creativecommons\.org/licenses/by/4\.0/](https://creativecommons.org/licenses/by/4.0/)\)\. Image is a cropped version of the original\. b\) Channel dimensions\. c\) Simulated results using our geometry\-free lift force model\.![Refer to caption](https://arxiv.org/html/2605.08109v1/spiral.png)Figure S8:Inertial separation of particles in spiral microchannels\. a\) Experimental results using\\qty6\\microand\\qty15\\microstyrofoam beads\. Adapted fromWarkianiet al\.[40](https://arxiv.org/html/2605.08109#bib.bib37)with permission from Springer Nature\. Scale bars are\\qty2\\centi\. b\) Simulated results using our geometry\-free lift force model\. Flow rates were\\qty100\\micro\\per\.