Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields

• URL: http://arxiv.org/abs/2507.17582v2
• Date: Wed, 17 Sep 2025 15:25:52 GMT
• Title: Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields
• Authors: Adrian Padilla-Segarra, Pascal Noble, Olivier Roustant, Éric Savin,
• Abstract summary: We present a general method for constraining a prescribed Gaussian process on an arbitrary compact set.<n>We derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles.<n>The relevance of the methodology and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile.
• Score: 0.6999740786886536
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Gaussian process regression techniques have been used in fluid mechanics for the reconstruction of flow fields from a reduction-of-dimension perspective. A main ingredient in this setting is the construction of adapted covariance functions, or kernels, to obtain such estimates. In this paper, we present a general method for constraining a prescribed Gaussian process on an arbitrary compact set. The kernel of the pre-defined process must be at least continuous and may include other information about the studied phenomenon. This general boundary-constraining framework can be implemented with high flexibility for a broad range of engineering applications. From this, we derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles. These kernels allow to define Gaussian process priors satisfying the incompressibility condition and the prescribed boundary conditions along the profile in a continuous manner. We describe an adapted numerical method for the boundary-constraining procedure parameterized by a measure on the compact set. The relevance of the methodology and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile, for which no observation at the boundary is needed at all.

Related papers

• ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance [35.08166384258028]
  ProFlow is a framework for zero-shot physics-consistent sampling.<n>It reconciles strict physical consistency and observational fidelity with the statistical structure of the pre-trained prior.<n>It achieves superior physical and observational consistency, as well as more accurate distributional statistics.
  arXiv  Detail & Related papers  (2026-01-28T03:57:00Z)
• Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints [0.6990493129893112]
  Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs)<n>Existing methods often rely on soft penalties or architectural biases that fail to guarantee hard constraints.<n>We propose Physics-Constrained Flow Matching, a zero-shot inference framework that enforces arbitrary nonlinear constraints in pretrained flow-based generative models.
  arXiv  Detail & Related papers  (2025-06-04T17:12:37Z)
• Physics-informed neural networks for hidden boundary detection and flow field reconstruction [0.0]
  This study presents a physics-informed neural network (PINN) framework designed to infer the presence, shape, and motion of static or moving solid boundaries.<n>The framework is validated across diverse scenarios, including incompressible Navier-Stokes and compressible Euler flows.<n>The proposed method exhibits robustness and versatility, highlighting its potential for applications when only limited experimental or numerical data are available.
  arXiv  Detail & Related papers  (2025-03-31T13:30:46Z)
• A Variational Computational-based Framework for Unsteady Incompressible Flows [0.0]
  We propose an alternative computational framework that employs variational methods to solve fluid mechanics problems.<n>First, it circumvents the chronic issues of pressure-velocity coupling in incompressible flows.<n>Second, this method eliminates the reliance on unphysical assumptions at the outflow boundary.
  arXiv  Detail & Related papers  (2024-12-07T03:45:39Z)
• Lattice real-time simulations with learned optimal kernels [49.1574468325115]
  We present a simulation strategy for the real-time dynamics of quantum fields inspired by reinforcement learning. It builds on the complex Langevin approach, which it amends with system specific prior information.
  arXiv  Detail & Related papers  (2023-10-12T06:01:01Z)
• GAFlow: Incorporating Gaussian Attention into Optical Flow [62.646389181507764]
  We push Gaussian Attention (GA) into the optical flow models to accentuate local properties during representation learning. We introduce a novel Gaussian-Constrained Layer (GCL) which can be easily plugged into existing Transformer blocks. For reliable motion analysis, we provide a new Gaussian-Guided Attention Module (GGAM)
  arXiv  Detail & Related papers  (2023-09-28T07:46:01Z)
• Posterior Contraction Rates for Mat\'ern Gaussian Processes on Riemannian Manifolds [51.68005047958965]
  We show that intrinsic Gaussian processes can achieve better performance in practice. Our work shows that finer-grained analyses are needed to distinguish between different levels of data-efficiency.
  arXiv  Detail & Related papers  (2023-09-19T20:30:58Z)
• Flow Matching on General Geometries [43.252817099263744]
  We propose a simple yet powerful framework for training continuous normalizing flows on manifold geometries. We show that it is simulation-free on simple geometries, does not require divergence, and computes its target vector field in closed-form. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets.
  arXiv  Detail & Related papers  (2023-02-07T18:21:24Z)
• Aspects of scaling and scalability for flow-based sampling of lattice QCD [137.23107300589385]
  Recent applications of machine-learned normalizing flows to sampling in lattice field theory suggest that such methods may be able to mitigate critical slowing down and topological freezing. It remains to be determined whether they can be applied to state-of-the-art lattice quantum chromodynamics calculations.
  arXiv  Detail & Related papers  (2022-11-14T17:07:37Z)
• Isotropic Gaussian Processes on Finite Spaces of Graphs [71.26737403006778]
  We propose a principled way to define Gaussian process priors on various sets of unweighted graphs. We go further to consider sets of equivalence classes of unweighted graphs and define the appropriate versions of priors thereon. Inspired by applications in chemistry, we illustrate the proposed techniques on a real molecular property prediction task in the small data regime.
  arXiv  Detail & Related papers  (2022-11-03T10:18:17Z)
• Gaussian Processes and Statistical Decision-making in Non-Euclidean Spaces [96.53463532832939]
  We develop techniques for broadening the applicability of Gaussian processes. We introduce a wide class of efficient approximations built from this viewpoint. We develop a collection of Gaussian process models over non-Euclidean spaces.
  arXiv  Detail & Related papers  (2022-02-22T01:42:57Z)
• A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information [0.0]
  In addition to training data, available information is not in the form of a finite collection of points.<n>We propose and construct processes that unify, via reproducing kernel Hilbert space, the typical finite training data case.<n>We show existence of the proposed process and establish that it is the limit of a conventional GP conditioned on an increasing number of training points.
  arXiv  Detail & Related papers  (2022-01-26T15:58:08Z)
• Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels [108.60991563944351]
  We present a recipe for constructing gauge equivariant kernels, which induce vector-valued Gaussian processes coherent with geometry. We extend standard Gaussian process training methods, such as variational inference, to this setting.
  arXiv  Detail & Related papers  (2021-10-27T13:31:10Z)
• Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems [91.3755431537592]
  We consider a control system of the form $dot x = sum_i=1lF_i(x)u_i$, with linear dependence in the controls. We use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points.
  arXiv  Detail & Related papers  (2021-10-24T08:57:46Z)
• Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems [48.7576911714538]
  We extend the physics-informed conditional Karhunen-Lo'eve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large-scale problems.
  arXiv  Detail & Related papers  (2021-07-30T18:43:14Z)
• A Framework for Fluid Motion Estimation using a Constraint-Based Refinement Approach [0.0]
  We formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. We also observe a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow.
  arXiv  Detail & Related papers  (2020-11-24T18:23:39Z)
• Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics [16.563723810812807]
  We propose a novel learning-based framework, the Neural Vortex Method (NVM) NVM builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics. By embedding these two networks with a vorticity-to-velocity Poisson solver, we can predict the accurate fluid dynamics.
  arXiv  Detail & Related papers  (2020-06-07T15:12:25Z)
• A Near-Optimal Gradient Flow for Learning Neural Energy-Based Models [93.24030378630175]
  We propose a novel numerical scheme to optimize the gradient flows for learning energy-based models (EBMs) We derive a second-order Wasserstein gradient flow of the global relative entropy from Fokker-Planck equation. Compared with existing schemes, Wasserstein gradient flow is a smoother and near-optimal numerical scheme to approximate real data densities.
  arXiv  Detail & Related papers  (2019-10-31T02:26:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.