Physics-Based Distribution Alignment

• Physics-based distribution alignment is a methodology that integrates physical constraints and priors to robustly align probability distributions reflecting real-world physical processes.
• Key approaches, including Bidirectional DeepParticle, PIETRA, and STFlow, leverage optimal transport, uncertainty-aware losses, and flow matching to enhance model fidelity and mitigate distribution shift.
• Advanced frameworks using information-geometric optimization and latent-space correction provide scalable, efficient solutions for high-dimensional inference with reduced computational complexity.

Physics-based distribution alignment is a suite of methodologies that utilize physical models, constraints, and principles to align probability distributions in data-driven and simulation-based inference tasks. This paradigm provides robustness against distribution shift, leverages domain-specific priors, and ensures that learned mappings or generative models remain consistent with known physical laws. Approaches vary by application, ranging from optimal transport in low-dimensional particle systems to deep neural network correction layers, flow-matching schemes, information-geometric optimization, and uncertainty-aware loss functions. The following sections delineate foundational principles, representative algorithms, theoretical and empirical properties, as well as scalability considerations in the context of physics-based distribution alignment.

1. Foundational Concepts and Motivation

Physics-based distribution alignment arises where distributions reflect outcomes constrained or determined by underlying physical processes—such as particle dynamics, chemotaxis, signal propagation, or geophysical measurements. Unlike purely data-driven alignment, this class of methods injects explicit or implicit physics constraints at various stages: prior construction, transport-map parameterization, loss function definition, or architecture design.

Central goals include:

• Robust generalization under distribution shift (i.e., changing observational regimes or model parameters)
• Faithful reproduction of physically valid solutions when data alone is insufficient or misleading
• Computational efficiency via symmetry, prior structure, or reduced parameter space
• Reduced risk of mode collapse, spurious solutions, or failure in out-of-distribution (OOD) regimes

Explicit models may include stochastic differential equations, partial differential equation (PDE) discretizations, physical priors, or random-walk processes informed by domain observables. Implicit approaches may utilize physics-informed losses, parameterized statistical manifolds, or adapt architectures via physical parameter concatenation.

2. Optimal Transport and Bidirectional Map Learning

A prominent methodology is the Bidirectional DeepParticle (BDP) approach for learning physics-based transport maps between distributions (Zhang et al., 16 Apr 2025). This framework seeks push-forward maps $F_\theta: X \to Y$ such that $F_\theta\sharp\pi_0 \approx \pi_1$, with $\pi_0$ as the reference (e.g., uniform or Gaussian) and $\pi_1$ the empirical target distribution, typically arising from particle discretizations of physical models.

BDP incorporates:

• Bidirectional training: Jointly learns $F_\theta$ and its inverse $G_\phi$, enforcing $G_\phi(F_\theta(x)) \approx x$ and $F_\theta(G_\phi(y)) \approx y$ for cycle consistency.
• Discrete 2-Wasserstein loss: Utilizes the bi-stochastic minimization

$$\widehat{W}_2(F_\theta) = \left( \min_{\Gamma \in \Gamma^N} \frac{1}{N} \sum_{i,j} \| F_\theta(x_i) - y_j \|^2 \Gamma_{ij} \right)^{1/2},$$

and similarly for $G_\phi$.

• Physics parameter conditioning: Concatenates variables such as chemotactic rate or flow amplitude to network inputs, aligning learned solutions with physical variability.
• Empirical efficacy: On 3D Keller–Segel and Kolmogorov flows, BDP achieves sub-$10^{-2}$ $W_2$ errors, outperforming rectified-flow and shortcut diffusion models; for dimensionality $d \geq 4$, single-step diffusion models surpass BDP due to $O(N^2)$ scalability limits.

Cycle consistency and dual loss terms substantially suppress mode collapse and spurious mappings prevalent in one-sided transport training. The principal challenge in high dimensions is the quadratic scaling in both memory and computation due to the $N \times N$ optimal transport coupling.

3. Physics-Informed Evidential Learning

The PIETRA framework exemplifies physics-informed distribution alignment for traversability estimation under distribution shift (Cai et al., 2024). Here, the approach integrates physical priors directly into the evidential neural architecture and the uncertainty-aware loss function.

Key components:

• Dirichlet posterior parameterization: For each input feature $x$, combines learned evidence $n_\lambda(x)$ and physical prior evidence $n_\text{phys}$:

$$\beta_b(x) = n_\text{phys} \, p_\text{physics}(b \mid x) + n_\lambda(x) \, p_\phi(b \mid x), \;\; b = 1, ... , B.$$

• Uncertainty-adaptive prediction: The predictive PMF leverages either the learned or physics prior distribution via adaptive weighting:

$$p_\theta(y \mid x) = \frac{n_\text{phys} p_\text{physics}(y \mid x) + n_\lambda(x) p_\phi(y \mid x)}{n_\text{phys} + n_\lambda(x)}.$$

• Uncertainty-aware physics-informed (UPI) loss: Loss function combines data fidelity and physics-model alignment terms, each computed as expectations over the Dirichlet posterior:

$$L^\text{UPI}(q_x; y, p_\text{physics}) = \mathbb{E}_{p \sim q_x} [\text{EMD}^2(p, y)] + \kappa \, \mathbb{E}_{p \sim q_x} [\text{EMD}^2(p, p_\text{physics}(x))].$$

• Adaptive OOD handling: Network transitions smoothly between learned and physics-based predictions depending on epistemic uncertainty, obviating the need for explicit gating.

PIETRA's experimental results demonstrate reduced Earth Mover's Distance errors and higher navigation success rates in OOD environments, with best alignment observed when explicit prior and implicit loss terms are jointly used.

4. Physics-Informed Priors in Generative Modeling

In probabilistic simulation of N-body and trajectory data, flow-matching and data-dependent coupling provide scalable physics-based distribution alignment (Brinke et al., 24 May 2025). The STFlow methodology establishes:

• Flow matching via continuity equation: Interpolates between prior $p_0$ (with physics-informed structure) and target $p_1$ using learned vector fields solving

$$\partial_t p_t(x) + \nabla_x \cdot (p_t(x) v_t(x)) = 0,$$

guaranteeing transport along physically plausible paths.

• Data-dependent coupling: Initializes within the conditioning frames, with random-walk prior noise $\zeta$ on unobserved frames respecting continuity, mean velocity, and variance estimated from observation.
• ODE-based trajectory synthesis: Sampling proceeds by drawing $x_0$ from prior, integrating ODE defined by learned flow field to target domain.
• Physical prior regularization: No additional loss terms are needed beyond flow matching, as the prior itself enforces physical constraints.

Empirically, STFlow delivers improved Average Displacement Error (ADE) and Final Displacement Error (FDE) versus prior state-of-the-art models, with order-of-magnitude reduction in inference steps due to reduced transport cost and straighter transformation paths.

5. Information-Geometric Optimization on Statistical Manifolds

Physics-based distribution alignment can be formalized on low-dimensional statistical manifolds derived from physics-constrained probability distributions (Boso et al., 2021). The Data-Assisted Method of Distributions (DA-MD) exploits the geometry of these manifolds for accelerated parameter inference.

Distinct features:

• Method of Distributions (MD): Solves closed-form PDEs for the probability law (CDF $F(X; t, \phi)$ or PDF $f$) as a function of system parameters $\phi$.
• Alignment via discrepancy metrics: Minimizes divergence (KL or Wasserstein) between physics-based prior $f(X; \phi)$ and Bayesian-updated posterior $\hat{f}(X)$,

$$D_{KL}(f \| \hat{f}) = \int f(X) \ln \frac{f(X)}{\hat{f}(X)} \, dX,$$

$$W_2^2(f, \hat{f}) = \int_0^1 |F^{-1}(u) - \hat{F}^{-1}(u)|^2 du.$$

• Natural gradient descent (NGD): Updates parameter estimates according to the local metric tensor (Fisher information for KL, Wasserstein for $W_2$),

$$\phi_{k+1} = \phi_k - \eta G^{-1}(\phi_k) \nabla_\phi \mathcal{C}(\phi_k).$$

• Deep NN surrogates: Trained networks approximate the MD mapping, enabling efficient calculation of PDFs, CDFs, quantiles, and parameter gradients via autodiff.

Comparative experiments show order-of-magnitude reductions in iterations when using NGD with manifold-informed preconditioning, robust recovery of target parameters, and enhanced tolerance to prior misspecification.

6. Latent-Space Correction for Robust Variational Inference

For inverse problems with distribution shift, physics-based latent distribution correction improves the fidelity of amortized variational inference (Siahkoohi et al., 2022).

• Latent prior relaxation: Instead of assuming standard normal latent variables in conditional flows, correction fits a Gaussian $\mathcal{N}(\mu, \Sigma)$ with diagonal covariance, optimized for each data instance.
• KL divergence minimization: $\mu^*, s^*$ found by minimizing reverse KL between the corrected latent and true physics-informed posterior,

$$(\mu^*, s^*) = \arg\min_{\mu, s} D_{KL}\left[q_\phi(z \mid y; \mu, \Sigma) \| p(z \mid y)\right].$$

• Stochastic variational optimization: Uses samples from the fitted latent prior to backpropagate through inverse flow and physics operator, enabling rapid adaptation.
• Sample quality and uncertainty quantification: Correction restores posterior contraction ratios, improves SNR by 2–4 dB in challenging settings, and provides precise credible intervals.

The correction operates at the cost of a handful of reverse-time migrations, in contrast to full non-amortized VI which carries orders-of-magnitude higher computational cost.

7. Scalability, Dimensionality, and Future Directions

Physics-based distribution alignment faces unique scalability bottlenecks. Quadratic complexity in sample-based optimal transport (as in BDP) limits utility to moderate dimensionality, while architectures employing flow-matching, information-geometric descent, or deep surrogates ameliorate these issues for higher-dimensional or structured data.

Potential remedy strategies include:

• Entropic OT solvers (e.g., Sinkhorn) and low-rank coupling approximations
• Sliced-Wasserstein or random-projection alignment for high-dimensional supports
• Hierarchical coupling or multiscale matching
• Physics-informed neural operators that integrate known PDE structure to reduce empirical sample complexity

Across domains—bio-physical modeling, wireless communication, robotic navigation, geophysical inference, and molecular dynamics—physics-based distribution alignment continues to provide robustness, interpretability, and efficiency not achievable with generic deep learning methods alone. The unifying theme is alignment that is grounded in, and varies adaptively with, the underlying physics of the data-generating process.