Physics-Based Flow Matching (PBFM)

• Physics-Based Flow Matching (PBFM) is a generative modeling framework that enforces physical constraints from PDEs and algebraic relations.
• It uses techniques like temporal unrolling and conflict-free multi-objective optimization to balance data fidelity with physical consistency.
• PBFM has demonstrated improved surrogate modeling and accelerated simulation performance in complex scenarios such as Darcy and Kolmogorov flows.

Physics-Based Flow Matching (PBFM) is a principled generative modeling framework that extends flow matching techniques with explicit incorporation of physical constraints, particularly those arising from partial differential equations (PDEs) and algebraic relations. By enforcing such constraints during generation, PBFM produces samples that are consistent both with observed data distributions and the underlying governing physics, making it highly suitable for surrogate modeling, uncertainty quantification, and accelerated physics simulation in scientific and engineering domains.

1. Definition and Rationale

PBFM integrates flow matching—a class of generative models based on transporting a base distribution to a target distribution along the trajectories of an ordinary differential equation (ODE)—with explicit enforcement of physical constraints such as PDE residuals. The objective is to efficiently generate samples that not only capture the data manifold but also satisfy the physical laws governing the system. Unlike traditional flow matching, which prioritizes data distribution matching, PBFM introduces physics-based residual losses directly into the objective, ensuring generated solutions are physically meaningful.

The core PBFM loss combines a flow matching term and a physics residual term: $$\mathcal{L} = w_{\mathrm{FM}}\,\mathcal{L}_{\mathrm{FM}} + w_\mathcal{R}\,\mathcal{L}_\mathcal{R}$$ where $\mathcal{L}_{\mathrm{FM}}$ is the standard flow matching loss and $\mathcal{L}_\mathcal{R}$ penalizes violations of PDE or algebraic constraints at the generated (final or intermediate) outputs.

2. Incorporation of Physical Constraints

PBFM embeds physical knowledge through direct minimization of the physical residual:

• For PDEs (e.g., Darcy flow), $\mathcal{R}(x)$ is the discretized residual of the governing equation, such as

$\mathcal{R} = \nabla \cdot (K\nabla p) + f = 0$

for steady-state flows in porous media.

• For analytic or algebraic relations (e.g., state equations), $\mathcal{R}(x)$ encodes exact constraints between physical fields.

This approach enables PBFM to explicitly control both distributional fidelity and physical consistency during model training and inference.

3. Temporal Unrolling for Accurate Residual Evaluation

Temporal unrolling is a training technique in which the model, starting from an intermediate time $t$, performs multiple small ODE integration steps up to the final time $t=1$, using the model's current velocity predictions. This creates a more accurate approximation of the actual generative trajectory, resulting in improved evaluation and minimization of the physical residual at the final sample.

The sequence is:

• For each batch and sampled $t$, generate $x_t$
• Unroll via

$x_{i+1} = x_i + dt \cdot u_t^\theta(x_i, t_i)$

until $t=1$, then compute $\mathcal{R}(x_1)$

This approach especially benefits early time points, where naively projecting to the end state would otherwise result in high physical inconsistency.

4. Conflict-Free Multi-Objective Optimization

Balancing the generative loss and physics residual loss is nontrivial; manual hyperparameter tuning of the weights is often unstable and leads to suboptimal trade-offs. PBFM utilizes a conflict-free multi-objective optimization scheme based on the ConFIG algorithm, in which gradients are adaptive and orthogonalized to ensure simultaneous progress on both objectives: $$\mathbf{g}_{\mathrm{update}} = (\mathbf{g}_{\mathrm{FM}}^\top \mathbf{g}_v + \mathbf{g}_\mathcal{R}^\top \mathbf{g}_v) \mathbf{g}_v$$ where $\mathbf{g}_v$ is computed by normalizing and orthogonalizing the individual gradients of each loss.

This guarantees each update step leads to improvement in both distributional and physical objectives, eliminating the need for hyperparameter tuning of loss weights.

5. Handling of Noise and Stochastic Sampling

The minimum noise level $\sigma_{\min}$ is often introduced as a form of regularization in FM. In PBFM, the addition of noise sets a lower bound on the achievable accuracy of the physical residual. Empirical results suggest setting $\sigma_{\min}=0$ is most suitable for physics-constrained scenarios, as it avoids artificially inflating physical error.

During sampling (inference), PBFM supports an additional stochastic sampling strategy where, after advancing forward in time, the process occasionally takes a step back while resampling the noise. This improves solution diversity and can further reduce physical residuals without compromising distributional accuracy, especially as the number of generative steps increases.

6. Empirical Performance and Applications

PBFM has been benchmarked on problems including:

• Darcy Flow (steady-state PDE): Up to eightfold reduction in physical residual MAE compared to vanilla FM, with superior surrogate modeling accuracy.
• Kolmogorov Flow (time-dependent PDE): Significantly lower residuals for incompressibility, with good agreement on first and second-order field statistics.
• Dynamic Stall (complex, multivariate, real-world data): Consistent reduction in physical constraint violation across multiple physical fields (e.g., pressure, wall shear) while maintaining or improving coverage of dataset variability.
• Toy Problems: On tasks such as unit-circle constraint, PBFM achieves residuals close to machine precision.

General applications include uncertainty-aware surrogate modeling, accelerated simulation, and scientific machine learning where strict constraint satisfaction is mandatory.

7. Comparison to Standard Flow Matching and Related Methods

PBFM distinctly improves over standard FM and other physics-informed generative models on key metrics:

Feature	Traditional FM	PBFM
Physics-Constrained	✗	✓ explicit (via residual loss)
Loss Weight Tuning	Manual	None (adaptive, conflict-free)
Temporal Unrolling	✗	✓
Stochastic Sampler	Deterministic	✓ optional, reduces residuals
Distributional Coverage	✓	✓ (not sacrificed for physics)
Physical Residual/Fidelity	Moderate	State-of-the-art

8. Broader Impact and Future Prospects

PBFM provides a practical and theoretically principled avenue for surrogate modeling, uncertainty quantification, and rapid simulation in physics and engineering. The framework is broadly applicable to any generative scenario where governing equations or residual-form constraints are known, and demonstrates scalability from low- to high-dimensional, time-dependent, and multivariate physical systems. It thereby serves as an enabling technology for accelerated design, real-time simulation, and probabilistic reasoning in data- and physics-constrained environments.