pith. sign in

arxiv: 2604.08194 · v1 · submitted 2026-04-09 · 💻 cs.LG · cs.NA· math.NA

Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations

Finn Sommer , Vamika Rathi , Sebastian Goetschel , Daniel Ruprecht This is my paper

Pith reviewed 2026-05-10 18:15 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords Basset forceMaxey-Riley-Gatignol equationsuniversal differential equationsneural network approximationparticle motionhistory effectsordinary differential equations
0
0 comments X

The pith

Neural networks approximate the Basset force to turn particle motion equations into standard ODEs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper uses universal differential equations to approximate the Basset force integral in the Maxey-Riley-Gatignol equations with neural networks. The result is a system of ordinary differential equations that can be integrated with conventional solvers such as Runge-Kutta methods. The Basset force captures history effects from wakes and boundary layers that influence particle trajectories in fluids. Including it matters because it changes both the speed and the qualitative paths particles take, effects that are often omitted due to the difficulty of handling the integral term. If the approximation works, it allows accurate simulations without special numerical techniques for the history term.

Core claim

The central claim is that a neural network trained in the universal differential equation framework can serve as a surrogate for the Basset history force. This surrogate replaces the integral in the Maxey-Riley-Gatignol equations, converting the model into an ordinary differential equation system that standard time-stepping methods can solve directly.

What carries the argument

A neural network embedded in a universal differential equation that approximates the convolution integral of the Basset force based on the particle's velocity history.

If this is right

  • The approximated model can be solved efficiently using existing ODE integrators without custom history-term handling.
  • Particle simulations can incorporate the Basset force for more realistic trajectories in applications involving fluid-particle interactions.
  • Training the network on representative trajectories allows the approximation to generalize across different flow conditions within the trained regime.
  • The approach avoids the computational cost and memory demands of evaluating the full integral at each time step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar neural approximations could handle other integral or history-dependent terms in physical models, such as in viscoelastic flows or memory-dependent materials.
  • In engineering contexts like pollutant dispersion or microfluidic devices, this could lead to faster yet accurate predictions of particle behavior.
  • One could test the method's robustness by applying it to particles with varying densities or sizes beyond the original training data.

Load-bearing premise

A neural network can learn to reproduce the effect of the Basset integral on particle motion with sufficient accuracy across relevant conditions.

What would settle it

Solve the original MaRGE with the full Basset integral using a specialized method and compare the resulting particle positions and velocities to those from the neural approximation under the same conditions; large discrepancies would show the approximation fails.

Figures

Figures reproduced from arXiv: 2604.08194 by Daniel Ruprecht, Finn Sommer, Sebastian Goetschel, Vamika Rathi.

Figure 1
Figure 1. Figure 1: Quiver plots of the two fluid fields used in the numerical experiments. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The used LSTM-Architecture consists of a recursive LSTM-cell that utilizes [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Final training and validation losses at the end of training depending on the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Training and validation loss during training for a training set with 100 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training and validation loss for the experimental field. The LSTM retains an [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory components of the UDE approach with FNN and LSTM applied [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Clustering of particles at t = 10 in the vortex. Ignoring the Basset force leads to an overestimation of the sinking speed of particles and thus a qualitatively different clustering pattern. Both FNN and LSTM are able to produce the correct patterns. reference in the horizontal coordinates. They capture the oscillating behaviour, but are unable to match frequency or amplitude. The reason is the difference … view at source ↗
Figure 8
Figure 8. Figure 8: The three components of the Basset force in the reference simulation and [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Approximation of the Basset force with increased particle density and no [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Relative errors of the UDE models with the solution that ignores the history [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Trajectories produced by the UDE approximation and reference solution [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Example trajectory for an initial position from the test split into coordinates. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Clustering of the final positions for both networks. The blue dots represent [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Components of the Basset force integrated over time for the experimental [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The error over time plot for the experimental field. Both models keep the [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
read the original abstract

The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes approximating the nonlocal Basset history integral in the Maxey-Riley-Gatignol equations (MaRGE) by embedding a neural network inside a universal differential equation framework, thereby converting the integro-differential system into a closed set of ordinary differential equations that can be integrated with standard solvers such as Runge-Kutta methods.

Significance. If the learned surrogate reproduces the effect of the Basset term on particle trajectories with usable accuracy over the relevant range of Stokes numbers, Reynolds numbers, and flow conditions, the approach would remove a long-standing numerical obstacle and allow routine inclusion of history effects in Lagrangian particle simulations without ad-hoc truncation or expensive quadrature.

major comments (2)
  1. The manuscript supplies no numerical validation, error comparisons against full MaRGE solutions, training details, or test cases, making it impossible to assess whether the approximation supports the claim of usable accuracy (see abstract and results description).
  2. No explicit statement is given of the parameter ranges (Stokes number, particle Reynolds number, fluid viscosity) used to generate training trajectories, nor are out-of-distribution tests reported; because the Basset term is a convolution over the entire past, any NN surrogate is only as reliable as its training coverage and extrapolation properties.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript proposing the use of universal differential equations to approximate the Basset force in the Maxey-Riley-Gatignol equations. We agree that additional numerical validation and specification of training parameters are required to fully assess the method's performance and will incorporate these in the revised manuscript.

read point-by-point responses

  1. Referee: The manuscript supplies no numerical validation, error comparisons against full MaRGE solutions, training details, or test cases, making it impossible to assess whether the approximation supports the claim of usable accuracy (see abstract and results description).

    Authors: We concur with the referee that numerical validation is essential to support the claims made in the abstract. The original manuscript focused on the methodological proposal of embedding a neural network within the universal differential equation framework to approximate the Basset force. In the revised manuscript, we will add a dedicated results section with numerical experiments, including direct comparisons to full MaRGE integrations, quantitative error metrics, details on the neural network training procedure, and multiple test cases demonstrating the approximation's performance. revision: yes

  2. Referee: No explicit statement is given of the parameter ranges (Stokes number, particle Reynolds number, fluid viscosity) used to generate training trajectories, nor are out-of-distribution tests reported; because the Basset term is a convolution over the entire past, any NN surrogate is only as reliable as its training coverage and extrapolation properties.

    Authors: We appreciate this observation regarding the importance of specifying the training domain and assessing generalization. The manuscript did not include these details as it presented the conceptual framework. In the revision, we will explicitly state the ranges of Stokes numbers, Reynolds numbers, and viscosities used to generate the training data, and we will include out-of-distribution tests to evaluate the surrogate's reliability for the history-dependent Basset term. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the UDE-based approximation method

full rationale

The paper proposes replacing the Basset history integral in MaRGE with a neural network inside a universal differential equation framework, yielding an ODE system solvable by standard integrators. This is explicitly framed as an empirical approximation learned from trajectory data rather than a first-principles derivation. No step reduces by construction to its own inputs, no fitted quantity is relabeled as an independent prediction, and no load-bearing self-citation chain is invoked. The method's validity rests on external numerical tests against the original integral, which are independent of the training procedure itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the assumption that a neural network can serve as a faithful surrogate for the Basset integral within the differential equation system; no new physical entities are introduced, but the neural network weights act as fitted parameters.

free parameters (1)
  • Neural network weights and biases
    Parameters of the neural network approximating the Basset force are determined by training on data.
axioms (1)
  • domain assumption The Basset force history effect can be adequately represented by a neural network embedded in the ODE right-hand side.
    Invoked when replacing the integral term with the learned network.

pith-pipeline@v0.9.0 · 5443 in / 1398 out tokens · 65473 ms · 2026-05-10T18:15:35.011921+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Cfd modelling and simulations of atomization-based processes for production of drug particles: A review.International Journal of Pharmaceutics, 670:125204, 2025

    Mohamad Baassiri, Vivek Ranade, and Luis Padrela. Cfd modelling and simulations of atomization-based processes for production of drug particles: A review.International Journal of Pharmaceutics, 670:125204, 2025

    work page 2025

  2. [2]

    Huusom, Tue Rasmussen, and Krist V

    Jonas Bisgaard, Monica Muldbak, Sjef Cornelissen, Tannaz Tajsoleiman, Jakob K. Huusom, Tue Rasmussen, and Krist V. Gernaey. Flow-following sensor devices: A tool for bridging data and model predictions in large-scale fermentations.Computational and Structural Biotechnology Journal, 18:2908–2919, 2020

    work page 2020

  3. [3]

    Bombardelli, Andrea E

    Fabi´ an A. Bombardelli, Andrea E. Gonz´ alez, and Yarko I. Ni˜ no. Computation of the particle basset force with a fractional-derivative approach.Journal of Hydraulic Engineering, 134(10):1513– 1520, 2008

    work page 2008

  4. [4]

    On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex.Physics of Fluids, 16(5):1765–1776, 2004

    F Candelier, JR Angilella, and M Souhar. On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex.Physics of Fluids, 16(5):1765–1776, 2004

    work page 2004

  5. [5]

    Advection of inertial particles in the presence of the history force: Higher order numerical schemes.Journal of Computational Physics, 254:93–106, 2013

    Anton Daitche. Advection of inertial particles in the presence of the history force: Higher order numerical schemes.Journal of Computational Physics, 254:93–106, 2013

    work page 2013

  6. [6]

    On the role of the history force for inertial particles in turbulence.Journal of Fluid Mechanics, 782:567–593, 2015

    Anton Daitche. On the role of the history force for inertial particles in turbulence.Journal of Fluid Mechanics, 782:567–593, 2015

    work page 2015

  7. [7]

    Universal differential equations as a unifying modeling language for neuroscience.Frontiers in Computational Neuroscience, Volume 19 - 2025, 2025

    Ahmed El-Gazzar and Marcel van Gerven. Universal differential equations as a unifying modeling language for neuroscience.Frontiers in Computational Neuroscience, Volume 19 - 2025, 2025

    work page 2025

  8. [8]

    Gatignol

    R. Gatignol. The Faxen formulae for a rigid particle in an unsteady non-uniform Stokes flow. Journal De Mecanique Theorique Et Appliquee, 2(2):143–160, 1983

    work page 1983

  9. [9]

    Andrea Gonz´ alez, Fabi´ an Bombardelli, and Yarko Ni˜ no. Improving the prediction capability of numerical models for particle motion in water bodies.Proceedings of the Seventh International Conference on Hydroscience and Engineering, 2007

    work page 2007

  10. [10]

    Long short-term memory.Neural Computation, 9(8):1735–1780, 1997

    Sepp Hochreiter and J¨ urgen Schmidhuber. Long short-term memory.Neural Computation, 9(8):1735–1780, 1997

    work page 1997

  11. [11]

    Jal´ on-Rojas, D

    I. Jal´ on-Rojas, D. Sous, and V. Marieu. A wave-resolving two-dimensional vertical lagrangian approach to model microplastic transport in nearshore waters based on trackmpd 3.0. Geoscientific Model Development, 18(2):319–336, 2025

    work page 2025

  12. [12]

    Interpolations.jl: Fast, continuous interpolation of discrete datasets in Julia, 2026

    JuliaMath. Interpolations.jl: Fast, continuous interpolation of discrete datasets in Julia, 2026

    work page 2026

  13. [13]

    and Sathiyamurthy K

    Vijayaprabakaran K. and Sathiyamurthy K. Towards activation function search for long short- term model network: A differential evolution based approach.Journal of King Saud University - Computer and Information Sciences, 34(6, Part A):2637–2650, 2022

    work page 2022

  14. [14]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017

    work page 2017

  15. [15]

    Liu and Jorge Nocedal

    Dong C. Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization. Mathematical Programming, 45(1):503–528, 1989

    work page 1989

  16. [16]

    Scientific machine learning of flow resistance using universal shallow water equations with differentiable programming.Water Resources Research, 61(9):e2025WR040265, 2025

    Xiaofeng Liu and Yalan Song. Scientific machine learning of flow resistance using universal shallow water equations with differentiable programming.Water Resources Research, 61(9):e2025WR040265, 2025. e2025WR040265 2025WR040265

    work page 2025

  17. [17]

    Equation of motion for a small rigid sphere in a nonuniform flow.The Physics of Fluids, 26(4):883–889, 1983

    Martin R Maxey and James J Riley. Equation of motion for a small rigid sphere in a nonuniform flow.The Physics of Fluids, 26(4):883–889, 1983

    work page 1983

  18. [18]

    Velocity measurement of a settling sphere.The European Physical Journal B-Condensed Matter and Complex Systems, 18(2):343–352, 2000

    Nicolas Mordant and J-F Pinton. Velocity measurement of a settling sphere.The European Physical Journal B-Condensed Matter and Complex Systems, 18(2):343–352, 2000

    work page 2000

  19. [19]

    Computation of the Basset force: recent advances and environmental flow applications.Environmental Fluid Mechanics, 16(1):193–208, 2016

    Patricio A Moreno-Casas and Fabian A Bombardelli. Computation of the Basset force: recent advances and environmental flow applications.Environmental Fluid Mechanics, 16(1):193–208, 2016

    work page 2016

  20. [20]

    Lux: Explicit Parameterization of Deep Neural Networks in Julia, April 2023

    Avik Pal. Lux: Explicit Parameterization of Deep Neural Networks in Julia, April 2023. If you use this software, please cite it as below

    work page 2023

  21. [21]

    On Efficient Training & Inference of Neural Differential Equations, 2023

    Avik Pal. On Efficient Training & Inference of Neural Differential Equations, 2023

    work page 2023

  22. [22]

    Current state and open problems in universal 24 differential equations for systems biology.npj Systems Biology and Applications, 11(1):101, 2025

    Maren Philipps, Nina Schmid, and Jan Hasenauer. Current state and open problems in universal 24 differential equations for systems biology.npj Systems Biology and Applications, 11(1):101, 2025

    work page 2025

  23. [23]

    Ganga Prasath, Vishal Vasan, and Rama Govindarajan

    S. Ganga Prasath, Vishal Vasan, and Rama Govindarajan. Accurate solution method for the Maxey–Riley equation, and the effects of Basset history.Journal of Fluid Mechanics, 868:428– 460, 2019

    work page 2019

  24. [24]

    Universal differential equations for scientific machine learning, 2021

    Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. Universal differential equations for scientific machine learning, 2021

    work page 2021

  25. [25]

    DifferentialEquations.jl–a performant and feature-rich ecosystem for solving differential equations in Julia.Journal of Open Research Software, 5(1), 2017

    Christopher Rackauckas and Qing Nie. DifferentialEquations.jl–a performant and feature-rich ecosystem for solving differential equations in Julia.Journal of Open Research Software, 5(1), 2017

    work page 2017

  26. [26]

    Numerical modeling of inertial particles in three-dimensional fluid flow, 2026

    Vamika Rathi and Daniel Ruprecht. Numerical modeling of inertial particles in three-dimensional fluid flow, 2026. Submitted

    work page 2026

  27. [27]

    Shake-the-box: Lagrangian particle tracking at high particle image densities.Experiments in Fluids, 57(5):70, 2016

    Daniel Schanz, Sebastian Gesemann, and Andreas Schr¨ oder. Shake-the-box: Lagrangian particle tracking at high particle image densities.Experiments in Fluids, 57(5):70, 2016

    work page 2016

  28. [28]

    Deep learning in neural networks: An overview.Neural Networks, 61:85–117, 2015

    J¨ urgen Schmidhuber. Deep learning in neural networks: An overview.Neural Networks, 61:85–117, 2015

    work page 2015

  29. [29]

    Numerical methods for the maxey-riley-gatignol equations, April 2026

    Finn Sommer. Numerical methods for the maxey-riley-gatignol equations, April 2026. url: https://doi.org/10.5281/zenodo.19471021

    work page doi:10.5281/zenodo.19471021 2026

  30. [30]

    Udesanalysis, April 2026

    Finn Sommer. Udesanalysis, April 2026. url: https://doi.org/10.5281/zenodo.19455827

    work page doi:10.5281/zenodo.19455827 2026

  31. [31]

    Universal differential equations for the maxey- riley-gatignol equations, April 2026

    Finn Sommer. Universal differential equations for the maxey- riley-gatignol equations, April 2026. url: https://doi.org/10.5281/zenodo.19471023

    work page doi:10.5281/zenodo.19471023 2026

  32. [32]

    Universal differential equations for the maxey- riley-gatignol equations - data, April

    Finn Sommer. Universal differential equations for the maxey- riley-gatignol equations - data, April

    work page

  33. [33]

    url: https://doi.org/10.5281/zenodo.19454255

    work page doi:10.5281/zenodo.19454255

  34. [34]

    The basset term as a semiderivative.Applied Scientific Research, 45(3):283–285, 1988

    FB Tatom. The basset term as a semiderivative.Applied Scientific Research, 45(3):283–285, 1988

    work page 1988

  35. [35]

    Relevance of the basset history term for lagrangian particle dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(7), 2025

    Julio Urizarna-Carasa, Daniel Ruprecht, Alexandra von Kameke, and Kathrin Padberg- Gehle. Relevance of the basset history term for lagrangian particle dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(7), 2025

    work page 2025

  36. [36]

    Efficient numerical methods for the maxey-riley-gatignol equations with basset history term.Computer Physics Communications, 310:109502, 2025

    Julio Urizarna-Carasa, Leon Schlegel, and Daniel Ruprecht. Efficient numerical methods for the maxey-riley-gatignol equations with basset history term.Computer Physics Communications, 310:109502, 2025

    work page 2025

  37. [37]

    An efficient, second order method for the approximation of the Basset history force.Journal of Computational Physics, 230(4):1465–1478, 2011

    MAT Van Hinsberg, JHM ten Thije Boonkkamp, and Hans JH Clercx. An efficient, second order method for the approximation of the Basset history force.Journal of Computational Physics, 230(4):1465–1478, 2011

    work page 2011

  38. [38]

    Tchinda Vermande Paganel, Epee Fabrice Alban, Mezoue Adiang Cyrille, and Claude Valery Ngayihi Abbe. Cfd simulation of an industrial dust cyclone separator: A comparison with empirical models: The influence of the inlet velocity and the particle size on performance factors in situation of high concentration of particles.Journal of Engineering, 2024(1):559...

    work page 2024

  39. [39]

    Cunteng Wang, Jingcui Xu, Haoyu Zhai, Lok Kwan So, and Hai Guo. Mapping full-range infection transmission from speaking, coughing, and sneezing in indoor environments and its impact on social distancing.Journal of Hazardous Materials, 490:137782, 2025

    work page 2025

  40. [40]

    Computational study of three-dimensional lagrangian transport and mixing in a stirred tank reactor.Chemical Engineering Journal Advances, 14:100448, 2023

    Christian Weiland, Eike Steuwe, J¨ urgen Fitschen, Marko Hoffmann, Michael Schl¨ uter, Kathrin Padberg-Gehle, and Alexandra von Kameke. Computational study of three-dimensional lagrangian transport and mixing in a stirred tank reactor.Chemical Engineering Journal Advances, 14:100448, 2023

    work page 2023