[Paper Review] Frequency-compensated PINNs for Fluid-dynamic Design Problems

Tongtao Zhang, Biswadip Dey+3 more|arXiv (Cornell University)|Nov 3, 2020

Model Reduction and Neural Networks30 references6 citations

TL;DR

This paper proposes a frequency-compensated physics-informed neural network (PINN) that improves generalization in fluid-dynamic design problems by learning Fourier features (frequency and phase offset) as functions of geometry and boundary conditions. By incorporating these features into a PINN framework, the model achieves accurate, fast predictions of velocity and pressure fields across unseen time intervals and design conditions, significantly outperforming standard PINNs in temporal and spatial generalization.

ABSTRACT

Incompressible fluid flow around a cylinder is one of the classical problems in fluid-dynamics with strong relevance with many real-world engineering problems, for example, design of offshore structures or design of a pin-fin heat exchanger. Thus learning a high-accuracy surrogate for this problem can demonstrate the efficacy of a novel machine learning approach. In this work, we propose a physics-informed neural network (PINN) architecture for learning the relationship between simulation output and the underlying geometry and boundary conditions. In addition to using a physics-based regularization term, the proposed approach also exploits the underlying physics to learn a set of Fourier features, i.e. frequency and phase offset parameters, and then use them for predicting flow velocity and pressure over the spatio-temporal domain. We demonstrate this approach by predicting simulation results over out of range time interval and for novel design conditions. Our results show that incorporation of Fourier features improves the generalization performance over both temporal domain and design space.

Motivation & Objective

• To address the challenge of generalizing physics-informed neural networks (PINNs) beyond training data in fluid-dynamic design problems.
• To improve temporal and spatial generalization of PINNs for incompressible flow around a cylinder by encoding periodic dynamics governed by the Strouhal number.
• To develop a surrogate model that predicts flow fields accurately for novel geometry and boundary condition settings without retraining.
• To reduce reliance on expensive high-fidelity simulations by enabling fast, accurate inference via a learned Fourier feature mapping subnetwork.

Proposed method

• Introduces a Fourier Feature Mapping (FFM) subnetwork that learns frequency and phase offset parameters as functions of inlet velocity and cylinder size.
• Integrates the FFM subnetwork into a PINN architecture to map spatial-temporal coordinates and design parameters into a frequency-aware latent space.
• Uses physics-based regularization via the Navier-Stokes equations to enforce physical consistency in the loss function.
• Employs a weighted loss function combining data fidelity and PDE residual terms, with a regularization weight λ=0.001 to balance accuracy and physical consistency.
• Trains the model on sparse anchor points from FeniCS simulations, enabling fast inference without full simulation re-runs.
• Uses automatic differentiation to compute PDE residuals during training, ensuring the network learns solutions consistent with the underlying PDEs.

Experimental results

Research questions

• RQ1 Can a PINN framework with learned Fourier features generalize better across time intervals beyond the training span?
• RQ2 How does incorporating frequency and phase offset learning improve generalization for unseen geometry and boundary condition settings?
• RQ3 What is the impact of removing Fourier feature mapping on prediction accuracy and phase consistency in untrained flow regimes?
• RQ4 How does physics-based regularization compare to standard PINN training in terms of generalization and robustness?
• RQ5 Can the proposed framework replace high-fidelity solvers like FeniCS in design exploration by achieving comparable accuracy with orders of magnitude faster inference?

Key findings

• The proposed frequency-compensated PINN achieves accurate predictions for unseen time intervals (e.g., t=5.15), with only minor phase shifts despite no training data in that range.
• When tested on unseen geometry settings (e.g., u_inlet=0.8, d_y=0.11), the model generalizes well, preserving flow structures and artifacts, as shown in visualization with low MSE.
• Removing the FFM component results in significant prediction errors and failure to capture complex frequency and phase shifts, demonstrating its critical role in generalization.
• Ablation studies show that without PDE regularization (λ=0), the model overfits to training data, while strong regularization suppresses vorticity in unseen cases.
• The framework reduces simulation time from over 24,000 seconds (FeniCS) to under 15,000 seconds for equivalent data generation, with comparable accuracy and full spatial-temporal coverage.
• The model successfully predicts flow fields across both spatial and temporal domains beyond the training data span, demonstrating robust generalization in design space.