Physics-informed neural particle flow for the Bayesian update step

Domonkos Csuzdi; Tamás Bécsi; Olivér Törő

cs.LGFeb 26, 2026

Physics-informed neural particle flow for the Bayesian update step

Authors: Domonkos Csuzdi, Tamás Bécsi, Olivér Törő

Abstract

The Bayesian update step poses significant computational challenges in high-dimensional nonlinear estimation. While log-homotopy particle flow filters offer an alternative to stochastic sampling, existing formulations usually yield stiff differential equations. Conversely, existing deep learning approximations typically treat the update as a black-box task or rely on asymptotic relaxation, neglecting the exact geometric structure of the finite-horizon probability transport. In this work, we propose a physics-informed neural particle flow, which is an amortized inference framework. To construct the flow, we couple the log-homotopy trajectory of the prior to posterior density function with the continuity equation describing the density evolution. This derivation yields a governing partial differential equation (PDE), referred to as the master PDE. By embedding this PDE as a physical constraint into the loss function, we train a neural network to approximate the transport velocity field. This approach enables purely unsupervised training, eliminating the need for ground-truth posterior samples. We demonstrate that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity. Experimental validation on multimodal benchmarks and a challenging nonlinear scenario confirms better mode coverage and robustness compared to state-of-the-art baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Physics based Bayesian computation. Center: The estimation objective is a static target posterior and its discrete approximation via an empirical measure. Left: Stochastic relaxation, where the system asymptotically converges to a fluctuating equilibrium ( $t\to\infty$ ). The macroscopic and microscopic views are linked via the mean-field limit. Right: Deterministic transport, where the system evolves over a finite horizon ( $\lambda\in[0,1]$ ). The particle trajectories correspond to the characteristic curves of the macroscopic master PDE.

Feature	MCMC	LHPF	SVGD	NF / CNF	PFBR	PINPF (Ours)
Work principle	Ergodic theory	Log-homotopy	Variational inference	Measure transport	Meta-learning	Physics-informed learning
Dynamics	Stochastic Markov chain	ODE or SDE	Deterministic gradient flow	Deterministic map / ODE	Deterministic neural ODE	Deterministic neural ODE
Update logic	Accept / reject	Numeric integration	Gradient descent & kernelized repulsion	Apply map / integrate ODE	Forward pass of learned $f_{\theta}$	Forward pass of learned $f_{\theta}$
Complexity (per step)	$O(N\cdot D^{2})$ (likelihood)	$O(N\cdot D^{3})$ (matrix inversion)	$O(N^{2}\cdot D)$ (pairwise kernel)	$O(N\cdot D^{2}\cdot L)$ (network depth)	$O(N\cdot D^{2}\cdot L)$ (network depth)	$O(N\cdot D^{2}\cdot L)$ (network depth)
Integration horizon	$t\to\infty$ (asymptotic)	$\lambda\in[0,1]$ (stiff)	Iterative opt. (until convergence)	$1$ step (map) / $\int_{0}^{1}$	$t\to\infty$ (asymptotic)	$\lambda\in[0,1]$ (discrete steps)
Training	No	No	No	Yes	Yes	Yes

Table 1: Overview of particle-based Bayesian inference methods in dimension

D

using

N

particles and

L

network layers.

Parameter	Value
MLP Architecture
Hidden Layers	6
Hidden Dimension	64
Activation Function (each layer)	SiLU
Training
Num. particles	500

Table 2: Hyperparameter settings for training and inference of PINPF.

Method	ED	SWD	GPU Time [s]
PINPF (ours)	0.1150	0.4201	0.0442
annealed MCMC	0.0958	0.3854	3.8336
NSF	0.1416	0.4186	0.0499
SVGD	0.1997	0.5262	1.4170
Incompressible Flow	0.2338	0.6052	0.1257

Table 3: Performance results on the four-dimensional Gaussian mixture posterior problem. The mean values are reported over the 100 samples from the test dataset.

Figure 2: Corner plot for a representative test inference task, which is selected as the one whose quantitative performance is closest to the mean across all 100 test problems (Table 3 ). Posterior samples from our method, SVGD, and annealed MCMC are shown alongside the analytic reference. Each method generates $1500$ particles, and marginal and pairwise densities are estimated via kernel density estimation.

Figure 3: The effect of different adaptive step thresholds

\Delta L

and particle numbers

N

on the validation dataset in terms of SWD and computational time.

Method	ED	SWD	Time [s]
PINPF (ours)	0.4352	0.7045	0.2243
annealed MCMC	0.1427	0.4142	4.3955
SVGD	0.4521	0.7701	2.2110
Incompressible flow	0.4819	0.7329	0.3003

Table 4: Zero-shot generalization performance on the out-of-distribution (GMM prior) dataset. PINPF was not retrained.

Figure 4: The likelihood landscape of the TDOA measurement. The nonlinear measurement equation creates a hyperbolic high-probability ridge.

Method	ED	SWD	Time [s]
PINPF (ours)	0.0697	0.3238	0.0196
Incompressible flow	0.2415	1.0460	0.3767
Local Gaussian exact flow	0.1749	0.6155	0.0132
Mean Gaussian exact flow	0.2521	0.6277	0.0101
SVGD	0.1344	0.4084	0.3733
annealed MCMC	0.0036	0.1527	4.8788

Table 5: Performance summary on the TDOA task (averaged over 100 test samples).

Figure 5: Qualitative comparison on a sample with an informative prior. The analytic flows struggle to capture the “banana” shape (Mean Exact) or properly cover the variance (Incompressible), while the Neural Flow matches the posterior geometry accurately.

Figure 6: Comparison on a sample with a distant prior. The Gaussian-based approximations (Exact flows) fail due to linearization errors. The Neural Flow robustly transports the particles to the posterior.

Figure 7: Particle trajectories generated by PINPF for the sample in Fig. 6 . The color gradient represents the flow time

\lambda\in[0,1]

. The model learns a smooth transport map that compresses the variance along the informative direction of the measurement.

Parameter	Value
PINPF – MLP Architecture
Hidden Layers / Hidden Dim ( $D\!=\!10$ )	6 / 64
Hidden Layers / Hidden Dim ( $D\!=\!15$ )	8 / 128
Activation Function	SiLU
PINPF – Training
Training particles	500

Table 6: Hyperparameter settings for the high-dimensional nonlinear experiments.

Method	ED	SWD	Time [s]
PINPF (ours)	0.2041	0.3228	0.0531
Incompressible flow	0.9486	0.6837	0.5668
Local Gaussian exact flow	0.2683	0.7221	0.0836
Mean Gaussian exact flow	0.2488	0.3840	0.0639
SVGD	0.3394	0.4455	0.0790
annealed MCMC	0.0388	0.2781	1.5966

Table 7: Performance on the 10-dimensional nonlinear problem. Mean values are reported over 100 test tasks.

Method	ED	SWD	Time [s]
PINPF (ours)	0.2313	0.3188	0.0663
Incompressible flow	1.2942	0.7243	0.5613
Local Gaussian exact flow	0.3331	0.6597	0.1272
Mean Gaussian exact flow	0.3199	0.3814	0.0963
SVGD	0.3637	0.4182	0.0816
annealed MCMC	0.0504	0.2642	1.5995

Table 8: Performance on the 15-dimensional nonlinear problem. Mean values are reported over 100 test tasks.

$\Delta\lambda_{\text{train}}$	ED	SWD
0.002	0.2294	0.3389
0.010	0.2041	0.3228
0.050	0.2403	0.3519
0.100	0.6186	0.5214

Table 9: Effect of training pseudo-time step

\Delta\lambda

on inference quality (

D=10

N=2000

\Delta L=0.5

, 100 test tasks).

Additional input features beyond $(x,\lambda,z)$	ED	SWD
none	0.6181	0.5162
$\log h$	0.5105	0.4672
$\nabla\log p_{\lambda}$	0.4318	0.4328
$\nabla\log h$	0.2503	0.3714
$\nabla\log g$	0.6466	0.5169
$\nabla\log p_{\lambda},\log h,\nabla\log h\text{ (all)}$	0.2041	0.3228

Table 10: Feature ablation on the

10\,

D nonlinear model (

N=2000

\Delta L=0.5

, 100 test tasks).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Random samples drawn from the training dataset for the TDOA problem. The dashed black lines represent the Gaussian prior, and the solid red lines represent the true posterior density.

Figure 9: Randomly selected samples from the test set comparing the ground truth posterior and the proposed Neural Flow method. Black dashed: Prior distribution. Red solid: Ground truth posterior. Blue solid: Kernel Density Estimate (KDE) of particles generated by our Neural Flow model.

Figure 10: Cornerplots of randomly selected test samples of the four-dimensional Gaussian mixture posterior problem. Red solid: Ground truth posterior. Blue solid: kernel density estimate of particles generated by PINPF.

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Physics-informed neural particle flow (PINPF) learns a deterministic transport field that moves particles from a prior distribution toward a Bayesian posterior while enforcing the governing probability-evolution equation. However, the standard PINPF velocity model processes particles independently and therefore does not explicitly condition its transport decisions on the empirical particle population. This paper introduces population-aware PINPF (PA-PINPF), which augments each particle update with a permutation-invariant Deep Sets representation of the full particle set. We investigate two population encoders. PA-PINPF-State summarizes the particle states, whereas PA-PINPF-Feature summarizes the complete local physics-informed feature vectors, including particle position, pseudo-time, measurement information, likelihood values, and score information. The latter allows the population context to represent not only particle-cloud geometry, but also the population-level Bayesian transport geometry. The methods retain the original unsupervised physics-informed residual objective and require no ground-truth posterior samples during training. Experiments on range-measurement tasks and nonlinear time-difference-of-arrival posterior transport demonstrate that both population-aware variants improve over particle-wise PINPF, while feature-population encoding provides the strongest performance. These results show that population-level physics features provide useful global information for learned Bayesian particle transport.

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