[Paper Review] Frame-independent vector-cloud neural network for nonlocal constitutive modeling on arbitrary grids

Xu‐Hui Zhou, Jiequn Han+1 more|arXiv (Cornell University)|Mar 11, 2021

Model Reduction and Neural NetworksPhysics and Astronomy87 references27 citations

TL;DR

This paper proposes a frame-independent vector-cloud neural network for nonlocal constitutive modeling on arbitrary grids, using a symmetry-invariant architecture that ensures invariance to translation, rotation, and point ordering. The method achieves robust, data-driven closure modeling for PDEs on unstructured meshes, demonstrating high accuracy on scalar transport problems over periodic hill geometries with superior generalization and computational efficiency.

ABSTRACT

Constitutive models are widely used for modeling complex systems in science and engineering, where first-principle-based, well-resolved simulations are often prohibitively expensive. For example, in fluid dynamics, constitutive models are required to describe nonlocal, unresolved physics such as turbulence and laminar-turbulent transition. However, traditional constitutive models based on partial differential equations (PDEs) often lack robustness and are too rigid to accommodate diverse calibration datasets. We propose a frame-independent, nonlocal constitutive model based on a vector-cloud neural network that can be learned with data. The model predicts the closure variable at a point based on the flow information in its neighborhood. Such nonlocal information is represented by a group of points, each having a feature vector attached to it, and thus the input is referred to as vector cloud. The cloud is mapped to the closure variable through a frame-independent neural network, invariant both to coordinate translation and rotation and to the ordering of points in the cloud. As such, the network can deal with any number of arbitrarily arranged grid points and thus is suitable for unstructured meshes in fluid simulations. The merits of the proposed network are demonstrated for scalar transport PDEs on a family of parameterized periodic hill geometries. The vector-cloud neural network is a promising tool not only as nonlocal constitutive models and but also as general surrogate models for PDEs on irregular domains.

Motivation & Objective

• To develop a nonlocal, data-driven constitutive model that is robust to diverse calibration datasets and applicable to unstructured meshes.
• To ensure frame-independence (invariant to translation, rotation, and Galilean transformations) in neural network-based constitutive modeling for physical systems.
• To enable region-to-point mapping using a vector-cloud representation that captures nonlocal physics such as turbulence and convection effects.
• To overcome limitations of traditional PDE-based closure models, which are rigid and hard to calibrate across complex flow regimes.
• To create a general-purpose surrogate model for PDEs on irregular domains with linear computational scaling in neighborhood size.

Proposed method

• Represent nonlocal flow information as a vector cloud: a set of points with associated feature vectors (coordinates, velocity, scalar quantities) around a target point.
• Construct an invariant feature matrix D via a composition of translational/rotational invariant operations: G⊤QQ⊤G⋆, where G is an embedding of scalar features and Q is the input data matrix.
• Use an embedding network to map scalar features ci to m-dimensional vectors, ensuring invariance through shared, identical processing across points.
• Apply a fitting network (fully connected) to the invariant feature matrix D ∈ ℝ^{m×m′} to predict the closure variable τ at the target point.
• Enforce permutational invariance by symmetric aggregation (e.g., averaging over point indices), ensuring independence from point ordering in the cloud.
• Train the model using Adam optimizer with 2000 epochs, batch size 1024, and a learning rate decay strategy (factor 0.7 every 600 epochs).

Experimental results

Research questions

• RQ1 Can a deep learning model for constitutive modeling be made invariant to coordinate system transformations (translation, rotation, Galilean) while preserving physical consistency?
• RQ2 How can nonlocal, region-to-point mappings be effectively modeled in unstructured meshes using neural networks?
• RQ3 Can a data-driven, frame-invariant neural network generalize across diverse flow configurations and parameterized geometries?
• RQ4 What architectural components are necessary to ensure invariance to point ordering and coordinate system choice in vector-cloud representations?
• RQ5 To what extent does the proposed model outperform traditional PDE-based closure models in accuracy and robustness on complex, non-periodic flow domains?

Key findings

• The proposed vector-cloud neural network achieves frame-independence through a composition of invariant operations, ensuring invariance to translation, rotation, and point ordering.
• The model demonstrates high accuracy in predicting closure variables for scalar transport PDEs on a family of parameterized periodic hill geometries, even with varying grid quality and flow conditions.
• The method exhibits linear computational complexity with respect to the number of points in the cloud, enabling efficient inference on large neighborhoods.
• The architecture successfully learns nonlocal physics, such as convection and diffusion effects, by encoding spatial correlations through the invariant feature matrix D.
• The model generalizes well across different parameterized flow configurations, suggesting strong potential as a surrogate for PDEs on irregular domains.
• The training process converges stably with a learning rate schedule, achieving consistent performance across multiple runs and geometries.