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The Neural Particle Method -- An Updated Lagrangian Physics Informed Neural Network for Computational Fluid Dynamics
Henning Wessels, Christian Weißenfels, Peter Wriggers
TL;DR
Mesh-based CFD faces challenges with stabilization and free-surface representation, while some physics-informed data-driven methods require fixed lattices or leave momentum unconstrained. This paper presents NPM, an Updated Lagrangian neural method for incompressible inviscid Euler flow. It reports conservation, agreement with experiments, and stable computations on irregularly distributed points.
Problem
Mesh-based CFD requires stabilization for incompressibility and convective terms and special techniques for continuous free surfaces; a cited data-driven approach also requires fixed lattices and leaves momentum a black box.
Source: §1 · pp. 2–3Method
NPM uses neural networks for velocity and pressure in an Updated Lagrangian formulation of the incompressible inviscid Euler equations, with high-order IRK integration and exact boundary conditions.
Source: §1 · p. 4, §5 · p. 17Results
NPM demonstrates excellent conservation, stable computations on irregular points, and dam-break fluid-front evolution in excellent agreement with reported experiments.
Source: Highlights · p. 1, §4.2 · p. 17Takeaways & Limitations
The method is reported to fulfill incompressibility without stabilization or topological restrictions on discretization points.
Source: §1 · p. 4The admissible time step remains limited because the Updated Lagrangian deformation map is linear.
Source: §5 · p. 17
Abstract
from arXiv · showhide
Numerical simulation is indispensable in industrial design processes. It can replace expensive experiments and even reduce the need for prototypes. While products designed with the aid of numerical simulation undergo continuous improvement, this must also be true for numerical simulation itself. Up to date, no general purpose numerical method is available which can accurately resolve a variety of physics ranging from fluid to solid mechanics including large deformations and free surface flow phenomena. These complex multi-physics problems occur for example in Additive Manufacturing processes. In this sense, the recent developments in Machine Learning display promise for numerical simulation. It has recently been shown that instead of solving a system of equations as in standard numerical methods, a neural network can be trained solely based on initial and boundary conditions. Neural networks are smooth, differentiable functions that can be used as a global ansatz for Partial Differential Equations (PDEs). While this idea dates back to more than 20 years ago [Lagaris et al., 1998], it is only recently that an approach for the solution of time dependent problems has been developed [Raissi et al., 2019]. With the latter, implicit Runge Kutta schemes with unprecedented high order have been constructed to solve scalar-valued PDEs. We build on the aforementioned work in order to develop an Updated Lagrangian method for the solution of incompressible free surface flow subject to the inviscid Euler equations. The method is easy to implement and does not require any specific algorithmic treatment which is usually necessary to accurately resolve the incompressibility constraint. Due to its meshfree character, we will name it the Neural Particle Method (NPM). It will be demonstrated that the NPM remains stable and accurate even if the location of discretization points is highly irregular.
Highlights
The highlights present a neural-network global ansatz with high-order IRK integration as the basis of the Neural Particle Method. They emphasize incompressibility without special treatment, conservation, and stability on irregular points.
Highlights
- A feed-forward neural network supplies a global geometric ansatz, with high-order implicit Runge Kutta integration for time.Source: Highlights · p. 1
- The method requires no special treatment of the incompressibility constraint.The highlights present this as a property of the approach.Source: Highlights · p. 1
- The method demonstrates excellent conservation properties in numerical examples.Source: Highlights · p. 1
- Computations remain stable even when discretization points are highly irregularly distributed.Source: Highlights · p. 1
1 Introduction
The introduction motivates a meshfree, physics-informed neural approach to incompressible free-surface flow, then presents the Neural Particle Method as an Updated Lagrangian solution of the inviscid Euler equations.
1 Introduction
- Standard mesh-based CFD faces stabilization needs for convective terms and incompressibility, and requires special techniques to represent continuous free surfaces.The introduction identifies these as three shortcomings of mesh-based methods for incompressible flow.Source: §1 · p. 2
- Meshfree particle methods follow fluid motion and naturally represent free surfaces, but commonly require connectivity established under topological requirements.ALE advances the mesh independently of fluid motion to avoid critical mesh distortion.Source: §1 · p. 2
- The introduction surveys machine-learning applications in CFD, including flow-field reconstruction, turbulence simulation, and learned latent-space dynamics.It notes data-driven approaches are increasingly used in CFD.Source: §1 · p. 3
- A physics-informed data-driven approach using convolutional stencils requires a fixed lattice and leaves the momentum equation a black box.The cited approach includes only the time-independent continuity equation as a physical constraint.Source: §1 · p. 3
- Neural networks can serve as global PDE ansatz functions because their input derivatives provide spatial derivatives, computed using automatic differentiation.The introduction describes the ansatz approach as a collocation method [Lagaris et al., 1998].Source: §1 · pp. 3–4
- Prior neural PDE work extended from static problems to time-dependent problems by treating IRK stages and the final solution as network outputs [Raissi et al., 2019].Earlier work also addressed complex domains and quasi-static mechanical problems [Lagaris et al., 2000; Samaniego et al., 2020; Nguyen-Thanh et al., 2020].Source: §1 · p. 4
- NPM applies an Updated Lagrangian formulation of the incompressible inviscid Euler equations, using neural networks for velocity and pressure with exact boundary conditions and high-order IRK.The method is presented as fulfilling incompressibility without stabilization on arbitrarily distributed points.Source: §1 · p. 4
2 A Physics Informed Neural Network for Second Order ODEs
The section explains how IRK integration is embedded in a neural network for second-order ODEs. Its oscillator example shows that high-order IRK can accurately span entire vibration cycles in large time steps.
2 A Physics Informed Neural Network for Second Order ODEs
- The section frames computational mechanics as solving a second-order equation of motion, with acceleration depending on velocity, position, and time.Source: §2 · p. 4
- IRK evaluates acceleration at stages between time steps; its methods are stable for stiff equations, and accuracy order can increase with the stage count.The stage coefficients are specified by a Butcher tableau.Source: §2 · pp. 4–5
- The network takes a discretization point's position as input and outputs its IRK velocity stages and next-step velocity; position updates follow from the velocity stages.The figure schematizes the position input and velocity outputs.Source: §2 · pp. 5–6, Figure 1 · p. 5
- Training minimizes a physics-informed loss built from velocity estimates, adjusting network weights and biases with Adam followed by L-BFGS-B.The loss is the sum of squared errors of the velocity estimates.Source: §2 · p. 6
- With s = 8 IRK stages, the network accurately integrates oscillator motion across time steps spanning entire oscillation cycles.The example uses one input neuron, two hidden layers of 20 neurons, and nine output neurons.Source: Figure 2 · p. 7, §2 · p. 7
3 Updated Lagrangian Formulation
The Updated Lagrangian formulation applies a neural network to the incompressible Euler equations, computing pressure stages so the incompressibility constraint is fulfilled. Boundary projection enforces Dirichlet conditions, while a penalty approach handles particle contact with rigid walls.
3 Updated Lagrangian Formulation
- The network takes positions at time tn and predicts IRK velocity stages, next-step velocity, and pressure stages for the incompressible Euler equations.Because spatial derivatives are formed with respect to the prior configuration, the method transforms them to the current configuration using the incremental deformation gradient.Source: §3 · pp. 7–8
- The incremental deformation gradient is integrated at each IRK stage from its rate, enabling stage-wise evaluation of the Euler equations.The velocity stages supply the rate used to obtain the deformation gradient.Source: §3 · p. 8
- Pressure has no evolution equation in this formulation; pressure stages are computed so the incompressibility constraint is fulfilled, and current pressure is their weighted average.Source: §3 · p. 8
- The loss combines velocity integration, velocity divergence, and pressure boundary-condition errors, with the divergence term accounting for the mass equation.Boundary-condition contributions are summed over boundary points, while velocity and divergence contributions are summed over points.Source: §3 · p. 9
3.1 Imposition of Dirichlet boundaries
- The boundary projection builds the velocity ansatz from boundary data and a distance function, which need only be C1 continuous for the Euler equations' first-order spatial derivatives.For complex geometries, smooth extensions and distance functions can be computed with low-capacity neural networks; rectangular cases can use analytical functions.Source: §3.1 · pp. 9–10
- Boundary projection accurately recovers the exact static pressure on highly irregular particle layouts, unlike simultaneous PDE-and-boundary training, which fails to satisfy Dirichlet conditions.The comparison uses a container discretized by 900 particles after 50 time steps.Source: §3.1 · pp. 9–10, Figure 3 · p. 10
3.2 Contact with rigid walls
- For rigid-wall contact, a penalty approach acts as an artificial volumetric spring based on the signed gap and activates when a particle crosses the boundary.The activation function ranges from 0 for no contact to 1 for contact.Source: §3.2 · p. 11
4 Numerical Results
The NPM is evaluated on inviscid free-surface sloshing and dam-break flows, with favorable comparisons to prior numerical and experimental results. The sloshing tests also reveal time-step limits on energy conservation.
4.1 Inviscid free sloshing
- At Δt = 1 s, adding IRK stages and network complexity does not overcome the time-step restriction.The comparison varies network layouts, including their number of IRK stages.Source: §4.1 · p. 12, Figure 7 · p. 13
- Only Δt = 0.1 s conserves energy; at Δt = 1 s artificial damping rapidly decays amplitude, while pressure and potential energy vary by less than 1%.The network layout is held constant at 20 IRK stages for this time-step comparison.Source: §4.1 · p. 12, Figure 6 · p. 13, Figure 5 · p. 13
- Higher-order approximations remain out of scope because they require a meaningful definition of line increment dxn.Source: §4.1 · p. 14
- At a = 0.2 m, overlapping waves break, and the computed sloshing agrees well with Oñate et al. [2004].The simulation uses randomly distributed particles and a time step of 0.1 s.Source: §4.1 · pp. 14–15, Figure 8 · p. 14
4.2 Dam break
- Only water-column outflow is simulated; wall interaction is outside scope because it requires additional surface identification for the zero-pressure boundary condition.The paper notes that an alpha-shape technique is one possible way to identify the surface.Source: §4.2 · p. 15
- The dam-break front tip agrees excellently with Martin and Moyce [1952], and the collapsed-column shape with Koshizuka and Oka [1996], though wall wettability is reduced.The comparison uses 20 particles per unit L; refining to 25 particles per unit L did not affect results.Source: Figure 10 · p. 16, Figure 11 · p. 16, §4.2 · p. 17
5 Conclusion
The Neural Particle Method applies an Updated Lagrangian formulation of the inviscid incompressible Euler equations to free-surface flow. It demonstrates conservation in sloshing while retaining a timestep constraint tied to its linear deformation map.
5 Conclusion
- NPM computes pressure that accurately fulfills incompressibility without topological restrictions on discretization points, unlike methods that can fail for disordered particle configurations.The method is formulated for the inviscid, incompressible Euler equations in an Updated Lagrangian manner.Source: §5 · p. 17
- Conservation properties are demonstrated in a sloshing test case, and the method performs well even for large sloshing amplitudes.Source: §5 · p. 17
- Although high-order IRK methods are used, admissible timestep sizes are limited because the incremental deformation gradient is a linear operator.The conclusion identifies relaxing the timestep constraint from the linear deformation map as future work to exploit IRK integration more fully.Source: §5 · p. 17