- The paper presents a data-driven surrogate for the Basset force that transforms the Maxey-Riley-Gatignol equations into a tractable system of ODEs.
- It employs both feedforward and LSTM architectures to achieve up to two orders of magnitude reduction in positional and clustering errors compared to models neglecting history effects.
- The approach enables efficient simulation of inertial particle dynamics in complex flow fields while extending model generalization well beyond the training intervals.
Neural Network-Based Approximation of the Basset Force in the Maxey-Riley-Gatignol Equations
Introduction
The Maxey-Riley-Gatignol equations (MaRGE) are the foundational model for describing the dynamics of spherical inertial particles immersed in a fluid, encompassing essential effects such as buoyancy, drag, added mass, and the Basset (history) force. The Basset force, accounting for memory effects from wake formation and boundary layers, introduces a convolution integral over the particle's velocity history, transforming MaRGE into an implicit integro-differential system. This memory term severely complicates both theoretical analysis and practical computation, as it makes the force at the current time depend non-locally on all past states, and introduces a singular kernel. Due to these computational burdens, the Basset term is often neglected in practice, despite experimental and numerical evidence showing that this omission introduces both quantitative and qualitative errors in predicted particle dynamics.
Universal Differential Equations for History Force Modelling
This work proposes a data-driven paradigm for approximating the Basset force in MaRGE via universal differential equations (UDEs), specifically by substituting the history integral with a neural network (NN) surrogate. UDEs offer a flexible hybrid framework, allowing the replacement of unknown or intractable physical terms with trainable function approximators within the ODE itself. Here, both feedforward neural networks (FNNs) and LSTMs are evaluated. Substituting the history term with a NN transforms MaRGE into a tractable system of ODEs, solvable by standard adaptive ODE solvers, eliminating the need for bespoke quadrature schemes or storing full trajectory histories.
The inputs to the NN are selected as time, the current particle state (position and velocity), the derivative of this state, the fluid velocity and its material derivative, and the fluid velocity at the initial position. The NN outputs a three-dimensional vector corresponding to the Basset force components. Both architectures maintain similar parameter counts (~13k), enabling a direct comparison.
Simulation Design and Data Generation
Reference trajectories are generated by a high-order explicit Adams-Bashforth scheme combined with specialized quadrature for the history term. Two canonical flow fields are considered:
- A synthetic 3D vortex with time- and z-dependent angular frequency, which elucidates inertial particle behavior in analytically defined structured turbulence.
- An experimentally measured turbulent field from a lab-scale stirred tank reactor, reconstructed on an Eulerian grid via high-speed Lagrangian particle tracking and interpolation.
Training, validation, and test datasets are generated by random sampling of initial positions and integrating MaRGE (with the full Basset term) over both short and extended time intervals, allowing the evaluation of model generalization and error accumulation in both familiar and extrapolative regimes.
Networks are trained to minimize the mean squared deviation between the trajectory predicted by the UDE-integrator and the reference MaRGE solution over all time points. Training employs the Adam optimizer for rapid convergence followed by L-BFGS for fine-tuning.
Key empirical findings include:
- Vortex Field: With minimal training data (e.g., 10 trajectories), LSTM slightly outperforms FNN, highlighting the benefit of explicit memory for capturing history effects. For larger datasets (100 trajectories), both models achieve nearly identical and substantially reduced errors relative to neglecting the history term.
- Quantitative improvement: Maximum point-wise positional errors are reduced by two orders of magnitude over models omitting the Basset term; clustering errors are similarly reduced.
- Experimental Field: UDE-based surrogates also outperform Basset-free models by approximately one order of magnitude in trajectory and clustering errors. Both models show slightly higher mean errors and volatility due to sharper gradients and more complex underlying fields.
- Both FNN and LSTM provide visually and quantitatively accurate approximations of clustering and asymptotic sinking behavior, capturing key qualitative phenomena absent in Basset-free models.
Basset Force Reconstruction and Generalization
Analysis of the NN-generated surrogate Basset force demonstrates that vertical force components, which dominate sinking behavior, are accurately reproduced, while errors in horizontal components (which are orders of magnitude smaller) are less well matched—an anticipated effect due to loss function scaling. In scenarios where horizontal forces dominate (e.g., increased particle density, zero gravity), the NN reallocates capacity and achieves accurate horizontal reconstructions.
Temporal extrapolation demonstrates that UDE-based models generalize effectively well beyond the training interval (up to 6x in the vortex field for LSTMs, up to 3x in the experimental field). However, error growth eventually manifests, with LSTM models retaining accuracy longer than FNNs. Notably, UDE-based models always outperform Basset-free models within the generalization window, but can be surpassed for long-term extrapolation if training data is insufficient or flow complexity increases.
Theoretical and Practical Implications
By establishing the feasibility of accurately substituting the intractable Basset history integral with NN-based surrogates inside a universal DE framework, this paper demonstrates that:
- The computational complexity of accurately capturing inertial particle dynamics can be drastically reduced, as the system reduces to a standard ODE amenable to off-the-shelf high-performance integrators.
- Sufficiently trained UDEs capture both quantitative and qualitative history effects, including realistic clustering and asymptotic sinking rates, even in highly turbulent or experimentally derived flow fields.
- The architecture choice (FNN vs. LSTM) is only mildly consequential given sufficient data, but recurrent models may offer small gains, especially for smaller datasets or for extrapolating history effects.
This approach opens a scalable avenue for integrating history effects—previously intractable for high-dimensional transient problems—into particle-laden flow simulation, environmental transport modelling, and dispersed phase CFD in industrial applications. The method is immediately applicable for moderate Reynolds number dilute suspensions and in scenarios where full DNS-CFD coupling is prohibitive.
Future Directions
The use of UDEs to approximate non-local terms in complex PDEs suggests several avenues for further research:
- Extension to non-spherical, polydisperse, or deformable particles.
- Application to highly turbulent, multi-phase, or reactive flows.
- Explicit handling of uncertainty quantification in the NN surrogate, potentially via Bayesian extensions.
- Investigation of solver-approximation interactions, adaptive loss balancing, and architecture optimization for capturing disparate scale interactions.
Conclusion
Neural network-based UDE surrogates allow for efficient and accurate approximation of the Basset force in MaRGE, enabling the retention of critical history effects in inertial particle modeling with minimal computational overhead. The combination of data-driven surrogates and physical modeling in the UDE framework offers a robust solution for longstanding computational challenges in Lagrangian particle dynamics and paves the way for further integration of ML models in computational physics contexts.