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Physics-Informed Diffusion Models
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Proposes physics-informed diffusion models (PIDMs), which add a PDE-residual likelihood term to the diffusion training loss so generated samples are pushed toward satisfying governing equations, evaluated on Darcy flow and topology optimization.
Problem
Diffusion models trained only on data do not enforce governing PDE constraints on generated samples, which matters for scientific applications; post-hoc projection methods do not change the learned distribution.
Approach
Treat PDE residuals as virtual observables with a Gaussian likelihood and add the negative residual log-likelihood, evaluated on an estimated clean sample x0* (mean estimate or DDIM sample) at each timestep with time-dependent variance, to the standard diffusion loss; inference is unchanged.
Key results
- On Darcy flow, PIDM reduces residual error by around two orders of magnitude compared with the standard diffusion model.
- On topology optimization (in-distribution), PIDM achieves R_MAE 1.24e-3 vs 1.86e-3 for standard diffusion and 1.51e-3 for CoCoGen, with MDN %CE 0.06 and %VFE 2.25.
- PIDM with mean estimation is less prone to overfitting, with test data loss recovering to near the vanilla model's level.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the physics-informed residual loss with sample estimation | Diffusion model · Topology optimization in-distribution | R_MAE: 1.86e-3 -> 1.24e-3; 33% reduction | Helped |
| Add the physics-informed residual loss with sample estimation | Diffusion model · Topology optimization out-of-distribution | R_MAE: 1.97e-3 -> 1.29e-3; 35% reduction | Helped |
| Add the physics-informed residual loss | Diffusion model · Topology optimization in-distribution | MDN % CE: 0.06 vs 0.09 (PG-Diffusion); 0.06 vs 0.14 (CoCoGen) | Helped |
| Add the physics-informed residual loss | Diffusion model · Topology optimization out-of-distribution | % VFE: 2.80 -> 1.91 | Mixed |
| Use the physics-informed loss instead of the standard data-driven objective | Diffusion model · Darcy flow | Residual error: around two orders of magnitude | Helped |
| Use mean estimation instead of sample (DDIM) estimation (ablation) | PIDM · Darcy flow | Residual error | Helped |
| Add physics-informed loss to the standard objective | Diffusion model · Unit circle toy problem | Residual error | Helped |
| Train only on residual loss, omitting the data likelihood (ablation) | PIDM · Unit circle toy problem | Sample diversity | Hurt |
| Use an uninformative Gaussian prior instead of training data (ablation) | PIDM · Unit circle toy problem | Residual error | Mixed |
| Increase residual-loss weighting (increase c) (ablation) | PIDM sample estimation · Unit circle toy problem | Residual error | Helped |
| Increase residual-loss weighting too far (ablation) | PIDM sample estimation · Unit circle toy problem | Sample diversity | Hurt |
| Predict the mean or noise rather than the clean signal (ablation) | PIDM · Darcy flow | Residual error | Hurt |
Surprising findings
- The standard diffusion model achieves a lower compliance error than PIDM on topology optimization despite higher volume fraction error, so results are not fully comparable.
- A model trained only on the residual loss collapses to a single point on the constraint manifold, and a PIDM with an uninformative Gaussian prior and no data can still approximate the target distribution, though less accurately.
Limitations
- Restricted to regular grids and geometrically simple domains, since residuals are computed with finite differences or FE stencils on fixed pixel grids.
- Requires a scaling hyperparameter c chosen by parameter sweep, and trade-offs between data and residual losses.
- Residual evaluation of intermediate estimates is not fully consistent with the marginal distribution (Jensen gap).
Experimental setup
Datasets: Darcy flow dataset (self-generated, 10,000 train / 1,000 validation) topology optimization dataset from prior work (30,000 structures)
Benchmarks: Darcy flow test set topology optimization in-distribution and out-of-distribution test sets
Metrics: mean absolute residual error (R_MAE) test data loss median compliance error (MDN % CE) volume fraction error (% VFE)
Baselines: Standard diffusion PG-Diffusion CoCoGen TopoDiff-G DOM
Builds on: denoising diffusion probabilistic models physics-informed neural networks DDIM min-SNR weighting CoCoGen
Closest papers by meaning
- The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning (NeurIPS 2026 · similarity 0.89)
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- DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · similarity 0.89)
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