What does this research mean for the field?

A supervised formulation of denoising diffusion probabilistic models (DDPM) can approximate solution operators for parameter-dependent PDEs with accuracy comparable to Fourier Neural Operators, while uniquely providing uncertainty quantification and the ability to recover noise magnitude from corrupted data. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To develop a data-driven generative modeling framework based on diffusion models for approximating the solution operators of parameter-dependent partial differential equations.

August 22, 2026Quarterly of Applied MathematicsOpen Access

Generative diffusion learning for parametric partial differential equations

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Authors

TWTing WangBeijing Institute of Technology PPPetr PlecháčUniversity of Delaware JKJaroslaw KnapUnited States Army Combat Capabilities Development Command

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Implication

Computational study reveals accurate operator learning for parametric partial differential equations via conditional diffusion models, highlighting built-in uncertainty quantification.

Key Points

To develop a data-driven generative modeling framework based on diffusion models for approximating the solution operators of parameter-dependent partial differential equations.
Adapted denoising diffusion probabilistic models (DDPM) into a supervised setting to represent the PDE solution operator as a family of conditional distributions.
Incorporated probabilistic uncertainty quantification through confidence intervals and enabled direct learning from noisy training data.
Benchmarked performance against Fourier Neural Operators (FNO) on parametric PDE tasks.
Demonstrated PDE solution accuracy comparable to standard Fourier Neural Operators.
Successfully estimated prediction uncertainty intervals and recovered the noise magnitude from data sets corrupted by additive noise.

Cite This Study

Wang et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c7e1 https://doi.org/10.1090/qam/1742

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August 22, 2026Quarterly of Applied MathematicsOpen Access

Generative diffusion learning for parametric partial differential equations

View Full Paper

Ask AI

Bookmark

Authors

TWTing WangBeijing Institute of Technology PPPetr PlecháčUniversity of Delaware JKJaroslaw KnapUnited States Army Combat Capabilities Development Command

Discussion

Member takes

Implication

Computational study reveals accurate operator learning for parametric partial differential equations via conditional diffusion models, highlighting built-in uncertainty quantification.

Key Points

To develop a data-driven generative modeling framework based on diffusion models for approximating the solution operators of parameter-dependent partial differential equations.
Adapted denoising diffusion probabilistic models (DDPM) into a supervised setting to represent the PDE solution operator as a family of conditional distributions.
Incorporated probabilistic uncertainty quantification through confidence intervals and enabled direct learning from noisy training data.
Benchmarked performance against Fourier Neural Operators (FNO) on parametric PDE tasks.
Demonstrated PDE solution accuracy comparable to standard Fourier Neural Operators.
Successfully estimated prediction uncertainty intervals and recovered the noise magnitude from data sets corrupted by additive noise.

Cite This Study

Wang et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c7e1 https://doi.org/10.1090/qam/1742

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