Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations
A neural network approximation makes a notoriously expensive fluid-dynamics equation solvable with standard solvers — but real-world validation remains limited.
The Thesis
When engineers simulate tiny particles drifting through fluid — think aerosol droplets in air or microplastics in water — they use a set of equations called the Maxey-Riley-Gatignol equations (MaRGE). MaRGE contains a term called the Basset force, which captures how a particle's entire past trajectory influences the drag it feels right now. That memory effect makes the equations unusually expensive to solve, because the solver must look back through all of history at every time step. This paper proposes replacing that expensive history integral with a neural network, turning a hard integro-differential equation into a standard system of ordinary differential equations that any off-the-shelf solver can handle. The catch is that the approach is demonstrated mostly on synthetic problems, and the accuracy-versus-speed trade-off in realistic, turbulent flow fields has not been rigorously benchmarked.
Catalyst
The concept of 'universal differential equations' — where neural networks are embedded directly inside differential equation systems and trained to approximate unknown or expensive terms — has matured enough in the last two to three years to support this kind of substitution. Simultaneously, automatic differentiation libraries (software that computes exact gradients through arbitrary code) now make training such hybrid systems straightforward. These two developments together made it practical to attempt a learned approximation of the Basset force where earlier efforts would have required hand-crafted mathematical closures.
What's New
Prior approaches to the Basset force problem fell into two camps: either ignore the term entirely (accepting known accuracy losses in swarm and dispersion modeling), or use specialized quadrature schemes — numerical integration methods that keep a compressed representation of the particle's history. Those quadrature methods reduce but do not eliminate the memory cost, and they require careful hand-tuning for each flow regime. This paper replaces the history integral with a neural network trained using the universal differential equations framework, so the resulting system has no explicit memory requirement and can be solved with standard Runge-Kutta methods without any regime-specific tuning.
The Counter
The paper's central demonstration relies on synthetic or simplified flow cases, not the turbulent, high-Reynolds-number environments where the Basset force causes the most trouble in practice. Neural network approximations of physical terms are notoriously prone to distributional shift: a network trained on one flow regime can fail silently when particle trajectories look different. The authors have not shown that the approximation is stable across the full range of Stokes numbers — the dimensionless parameter that governs how strongly a particle responds to fluid motion versus its own inertia — that practitioners actually care about. Existing quadrature-based methods, while more expensive, come with error bounds; a learned term provides no such guarantee. Until the method is validated on canonical turbulent benchmarks and the accuracy-versus-speed numbers are published alongside error statistics, this is a promising proof of concept rather than a deployable technique.
Longs
- ANSYS (ANSS) — simulation software vendor whose solvers would integrate such methods
- Dassault Systèmes (DSY) — multiphysics simulation platform with direct fluid-particle modeling use cases
- MathWorks (private) — MATLAB/Simulink ecosystem where ODE solvers are the dominant toolchain
Shorts
- Vendors of specialized Basset-force quadrature libraries — their niche value proposition shrinks if a general learned approximation proves accurate
- Research groups with large investments in custom history-tracking numerical schemes — methodological sunk costs become less relevant
Enablers (Picks & Shovels)
- Julia SciML ecosystem (open-source) — the primary software environment for universal differential equations and neural ODE research
- JAX / PyTorch autograd — automatic differentiation backends that make training network-embedded ODE systems practical
- arXiv preprint culture in scientific ML — rapid dissemination means fluid-dynamics engineers can adopt and test quickly
Private Watchlist
- Pasteur Labs — applies differentiable simulation to industrial fluid modeling
- Flexcompute — specializes in fast PDE solvers for engineering applications
- Calculus AI / similar scientific ML startups in hybrid neural-ODE space
Resources
The Paper
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.