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A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction
Dule Shu, Zijie Li, Amir Barati Farimani
TL;DR
Existing reconstruction models can lose accuracy when test-time low-fidelity inputs differ from their training data, creating a need for retraining. This paper instead trains a diffusion framework on high-fidelity data, denoises noisy low-fidelity conditioning inputs, and optionally uses PDE information. The model achieves similar or marginally better L2 reconstruction accuracy, with greater robustness to low-fidelity input variation and more accurate kinetic energy spectra.
Problem
Existing CFD reconstruction models are trained for particular under-resolved inputs and can lose accuracy when test inputs differ substantially from the training data, requiring retraining for best performance.
Method
The paper trains a DDPM-based denoising framework using high-fidelity data and reconstructs high-fidelity CFD fields from noisy low-fidelity conditioning inputs, optionally incorporating PDE information.
Results
The model provides similar or marginally better reconstruction accuracy in L2 loss, is more robust to low-fidelity input variation, and is more accurate in kinetic energy spectrum.
Takeaways & Limitations
The diffusion framework supports high-fidelity CFD reconstruction across varying low-fidelity inputs without retraining.
Takeaways & Limitations
The conclusion identifies the use of an Lp reconstruction loss as a potential limitation of diffusion-model reconstruction.
Abstract
from arXiv · showhide
Machine learning models are gaining increasing popularity in the domain of fluid dynamics for their potential to accelerate the production of high-fidelity computational fluid dynamics data. However, many recently proposed machine learning models for high-fidelity data reconstruction require low-fidelity data for model training. Such requirement restrains the application performance of these models, since their data reconstruction accuracy would drop significantly if the low-fidelity input data used in model test has a large deviation from the training data. To overcome this restraint, we propose a diffusion model which only uses high-fidelity data at training. With different configurations, our model is able to reconstruct high-fidelity data from either a regular low-fidelity sample or a sparsely measured sample, and is also able to gain an accuracy increase by using physics-informed conditioning information from a known partial differential equation when that is available. Experimental results demonstrate that our model can produce accurate reconstruction results for 2d turbulent flows based on different input sources without retraining.
Introduction
The paper targets high-fidelity CFD reconstruction from low-fidelity inputs to reduce the conflict between computational cost and simulation fidelity. It proposes a diffusion-based framework designed to remain accurate when test inputs differ from training data, including with physics-informed conditioning.
- Motivation: High-fidelity CFD methods can be computationally expensive, motivating machine-learning surrogate models that reduce the cost of producing or reconstructing flow data.Low-fidelity simulations require fewer computational resources, while reconstruction aims to recover high-fidelity results and improve CFD cost-effectiveness.
- Limitation of prior work: Existing CFD reconstruction models are trained for particular under-resolved data types, such as a specific filter or resolution.Their training assumptions constrain the low-fidelity data distribution used by the model.
- Limitation of prior work: When test-time low-fidelity inputs substantially deviate from the training dataset, reconstruction accuracy drops significantly and retraining is generally required.The cited deviations include changes in resolution or Gaussian blurring processes.
- Proposed approach: The proposed framework reformulates high-fidelity reconstruction from low-fidelity input as denoising and uses a Denoising Diffusion Probabilistic Model.The diffusion model is trained using high-fidelity data and reconstructs high-fidelity CFD data from noisy input.
- Proposed approach: Physics-informed conditioning incorporates information from the governing PDE during diffusion-model training and sampling to improve reconstruction accuracy.The approach uses PDE-based conditioning information associated with the fluid flow.
- Results: Experiments on 2d turbulent flows show comparable performance to state-of-the-art models while preserving accuracy across different input distributions without retraining.The reported evaluation includes kinetic energy spectrum and PDE residual loss.
Method
The method converts low-fidelity CFD inputs into noisy conditioning samples and applies a partial backward diffusion process to reconstruct high-fidelity flow fields. Noise reduces distribution mismatch between training and test inputs, while DDPM denoising and optional guidance produce the reconstruction.
- Problem formulation: The method addresses train–test distribution mismatch by preprocessing low-fidelity inputs so their distributions become more similar before reconstruction.The preprocessing procedure g adds Gaussian noise, drawing the resulting distributions toward Gaussian distributions.
- Denoising Diffusion Probabilistic Model: During DDPM training, the network predicts noise from a noisy data sample and timestep within a forward diffusion process.The forward process progressively adds Gaussian noise, while the learned reverse process generates denoised data.
- Preprocessing: The preprocessing procedure g := h(x, z) mixes a low-fidelity sample x with random noise z to create a noisy conditioning sample.The resulting noisy samples are passed to a pretrained DDPM denoising module.
- Denoising Diffusion Probabilistic Model: The DDPM module performs partial backward diffusion from an intermediate conditioning sample xt = g(x) to a denoised output x0 = fθ(x).The output is intended to approximate the corresponding high-fidelity ground-truth sample.
- Guided data synthesis: The framework can start the reverse diffusion chain at an available intermediate timestep rather than from pure noise, enabling guided generation from low-fidelity references.The Markovian property permits generation from any timestep when the corresponding intermediate sample is available.
- Guided data synthesis: For conditional generation, the low-fidelity reference supplies conditioning information while noise mixing brings its intermediate representation closer to the high-fidelity training representation.This is intended to mitigate disturbance caused by the discrepancy between the reference and high-fidelity data.
Improved Procedures for DDPM-based Conditional Sampling
The framework modifies DDPM-based conditional sampling to reconstruct high-fidelity CFD data from low-fidelity or sparse inputs, optionally using PDE-derived guidance. It implements physics-informed guidance either through learned conditioning or direct gradient descent during sampling.
- Sparse measurements reduce available data and make reconstruction challenging because unmeasured components must be recovered.
- The iterative sampling procedure repeatedly refines reconstructions from sparsely measured data, using the previous reconstruction as a low-fidelity reference.
- Known PDEs provide physics-informed guidance through the gradient of the PDE residual computed from intermediate diffusion states.
- Learned residual guidance: The framework offers learned residual guidance, where a modified DDPM conditions sampling on c = ∂r_t/∂x_t.
- Direct gradient descent: A second method directly applies gradient descent during DDIM sampling without modifying the original DDPM architecture or training procedure.
- Conditional sampling combines data and PDE-residual-gradient predictions, with their relative contribution controlled by the guidance strength w.
Experiments
Experiments use simulated two-dimensional Kolmogorov flow data governed by a vorticity-form incompressible-flow equation. The study evaluates diffusion-based reconstruction strategies across upsampling and sparse-measurement settings, including physics-informed variants and fixed-map baselines.
- Dataset: High-fidelity data come from direct numerical simulation and serve as the ground-truth reconstruction targets.
- Dataset: The dataset contains 40 Kolmogorov-flow sequences, each lasting 10 seconds and comprising 320 frames after downsampling to a 256 × 256 grid.
- Dataset: The first 36 sequences are used for training and the remaining 4 sequences for testing.
- PDE residual evaluation: PDE residuals are computed using Fourier methods for spatial derivatives and finite differences for time derivatives.
- Compared models: The UNet-based study compares learned residual guidance, the original diffusion model, and a fixed low-fidelity-to-high-fidelity mapping.
- Compared models: The fixed mapping requires paired training data and is usually locked to a specific low-fidelity input distribution.
Results
The diffusion model reconstructs high-resolution flow fields across upsampling and sparse-observation tasks without retraining, producing detailed and physically coherent results. Physics-informed guidance further reduces PDE residuals, while diffusion remains more robust than direct mapping on out-of-distribution inputs.
- Upsampling: 4× upsampling results are satisfactory for both bicubic interpolation and diffusion, while diffusion recovers more detail in the 8× task.The visual comparisons cover 64 × 64 → 256 × 256 and 32 × 32 → 256 × 256 inputs.
- Quantitative comparison: Both diffusion and learned direct mapping outperform bicubic interpolation in pointwise error, while direct mapping degrades more on a new data distribution.Direct mapping has similar L2 performance to diffusion on its training distribution, but the gap increases for uniformly downsampled inputs.
- Quantitative comparison: Diffusion is more robust than direct mapping on sparse inputs, where direct mapping’s L2 and equation losses increase significantly.The direct mapping model is sensitive to input distribution, whereas diffusion retains stronger performance under distribution shift.
- Physics-informed guidance: Physics-informed guidance yields lower PDE residuals than models without PDE information, and nonlinear residual-gradient and score-function guidance converges better than linear combination.Diffusion also captures high-wavenumber energy trends with less truncation than bicubic interpolation.
- Statistical evaluation: Diffusion predictions show sharper vorticity distributions than ground truth and bicubic results, likely because noise-estimation error causes distribution drift.The drift becomes more evident at higher sparsity, where larger noise and more diffusion steps are used.
Conclusion
The work presents a diffusion model for high-fidelity CFD reconstruction from low-fidelity or sparsely measured inputs. It achieves comparable L2 accuracy to direct-mapping models while offering greater robustness to low-fidelity input variation and improved kinetic energy spectra, with physics-informed conditioning supporting improved performance.
- The proposed diffusion model reconstructs high-fidelity CFD data from low-fidelity inputs, including sparsely measured samples.The approach uses different configurations for low-fidelity and sparsely measured inputs.
- The model frames reconstruction as conditional data denoising rather than direct mapping from low-fidelity to high-fidelity data.Its training objective minimizes KL-divergence between forward and backward diffusion processes instead of directly minimizing an Lp reconstruction loss.
- The model has similar, marginally better L2 reconstruction accuracy than the benchmark direct-mapping method.
- The diffusion model is much more robust to variation in low-fidelity input data and more accurate in terms of data kinetic energy spectrum.
- Physics-informed information such as the PDE residual gradient can be incorporated during training and inference for improved performance.
- The absence of an Lp reconstruction loss is identified as a potential limitation, motivating new sampling designs or ensemble methods combining diffusion and direct-mapping models.These approaches are presented as directions for future work.