[Paper Review] Physics Informed Machine Learning of SPH: Machine Learning Lagrangian Turbulence.

Michael Woodward, Yifeng Tian+5 more|arXiv (Cornell University)|Oct 25, 2021

Model Reduction and Neural Networks49 references4 citations

TL;DR

This paper introduces a physics-informed machine learning framework for smoothed particle hydrodynamics (SPH) that integrates neural networks with physics-based parameters to model Lagrangian turbulence. By combining forward and reverse mode automatic differentiation with sensitivity analysis, the method enables efficient optimization, solving inverse problems, learning turbulence statistics, and generalizing beyond training data while preserving physical symmetries and reducing data requirements.

ABSTRACT

Smoothed particle hydrodynamics (SPH) is a mesh-free Lagrangian method for obtaining approximate numerical solutions of the equations of fluid dynamics; which has been widely applied to weakly- and strongly compressible turbulence in astrophysics and engineering applications. We present a learn-able hierarchy of parameterized and physics-explainable SPH informed fluid simulators using both physics based parameters and Neural Networks (NNs) as universal function approximators. Our learning algorithm develops a mixed mode approach, mixing forward and reverse mode automatic differentiation with forward and adjoint based sensitivity analyses to efficiently perform gradient based optimization. We show that our physics informed learning method is capable of: (a) solving inverse problems over the physically interpretable parameter space, as well as over the space of NN parameters; (b) learning Lagrangian statistics of turbulence (interpolation); (c) combining Lagrangian trajectory based, probabilistic, and Eulerian field based loss functions; and (d) extrapolating beyond training sets into more complex regimes of interest. Furthermore, this hierarchy of models gradually introduces more physical structure, which we show improves interpretability, generalizability (over larger ranges of time scales and Reynolds numbers), preservation of physical symmetries, and requires less training data.

Motivation & Objective

• To develop a learnable, physics-explainable hierarchy of SPH simulators that integrate neural networks with physical parameters for improved modeling of Lagrangian turbulence.
• To address the challenge of limited generalizability and interpretability in purely data-driven SPH models by embedding physical laws into the learning framework.
• To reduce data requirements and improve robustness across diverse Reynolds numbers and time scales through structured physical priors in the model architecture.
• To enable inverse problem solving over both physical parameters and neural network weights using gradient-based optimization.
• To combine multiple loss functions—Lagrangian trajectory-based, probabilistic, and Eulerian field-based—within a unified learning framework for comprehensive turbulence representation.

Proposed method

• The framework employs a mixed-mode learning approach using forward and reverse mode automatic differentiation to compute gradients for both physics-based parameters and neural network weights.
• Sensitivity analysis is performed using forward and adjoint methods to efficiently compute gradients required for optimization in high-dimensional parameter spaces.
• A hierarchical model structure gradually introduces physical constraints into the neural network, enhancing interpretability and preserving physical symmetries.
• The method supports multi-modal loss functions: Lagrangian trajectory tracking, probabilistic density estimation, and Eulerian field reconstruction, enabling comprehensive training signals.
• Neural networks act as universal function approximators within a physics-informed loss function, ensuring that predictions remain consistent with the underlying fluid dynamics equations.
• The optimization process is guided by a composite loss combining physical consistency, data fidelity, and statistical accuracy across different representations of the flow.

Experimental results

Research questions

• RQ1 Can a physics-informed machine learning framework effectively solve inverse problems in SPH by learning both physical parameters and neural network weights simultaneously?
• RQ2 To what extent can neural networks trained on SPH data generalize to unseen regimes of Reynolds number and time scale while preserving physical consistency?
• RQ3 How does embedding physical structure into the model hierarchy affect interpretability, data efficiency, and preservation of symmetries in Lagrangian turbulence simulations?
• RQ4 Can a unified loss function combining Lagrangian trajectories, probabilistic distributions, and Eulerian fields improve the accuracy and robustness of SPH simulations?
• RQ5 Does the integration of physics-based constraints in a learnable SPH framework lead to better extrapolation performance beyond the training distribution?

Key findings

• The method successfully solves inverse problems in both the physical parameter space and the neural network weight space using gradient-based optimization with high efficiency.
• The model demonstrates strong generalization capabilities, extrapolating to more complex turbulence regimes beyond the training data distribution.
• Incorporating physical structure into the model hierarchy improves interpretability and preserves key physical symmetries such as momentum and energy conservation.
• The framework requires less training data compared to purely data-driven approaches due to the inductive bias from physics-informed constraints.
• The combination of Lagrangian trajectory, probabilistic, and Eulerian field-based loss functions leads to more accurate and comprehensive representation of turbulence statistics.
• The use of mixed-mode automatic differentiation and adjoint sensitivity analysis enables scalable and efficient training on high-dimensional SPH simulation problems.