PFGM++: Unlocking the Potential of Physics-Inspired Generative Models (2302.04265v2)

Abstract: We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for $N$ dimensional data by embedding paths in $N{+}D$ dimensional space while still controlling the progression with a simple scalar norm of the $D$ additional variables. The new models reduce to PFGM when $D{=}1$ and to diffusion models when $D{\to}\infty$. The flexibility of choosing $D$ allows us to trade off robustness against rigidity as increasing $D$ results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of $D$, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models ($D{\to} \infty$) to any finite $D$ values. Our experiments show that models with finite $D$ can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ $64{\times}64$ datasets, with FID scores of $1.91/2.43$ when $D{=}2048/128$. In class-conditional setting, $D{=}2048$ yields current state-of-the-art FID of $1.74$ on CIFAR-10. In addition, we demonstrate that models with smaller $D$ exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

• The paper introduces PFGM++, a breakthrough model that unifies diffusion and PFGM by embedding N-dimensional data into an augmented N+D space.
• It employs a novel perturbation-based objective that removes large batch constraints and supports efficient conditional generation.
• Empirical results on CIFAR-10 and FFHQ demonstrate state-of-the-art FID scores, showcasing its flexible balance between robustness and rigidity.

PFGM++: A Unified Approach to Physics-Inspired Generative Models

The paper introduces a novel framework for generative modeling, expanding the existing concepts of diffusion models and Poisson Flow Generative Models (PFGM) into a comprehensive family named PFGM++. By embedding $N$-dimensional data trajectories into an $N{+}D$-dimensional space, this approach efficiently bridges the gap between diffusion models and PFGMs, unifying them through a physics-inspired lens.

Theoretical Insights and Model Formulation

PFGM++ adapts ideas from electrostatics, redefining traditional PFGM by introducing a $D$-dimensional augmentation. In this construct, diffusion models become a special case as $D$ approaches infinity, and PFGM is recovered when $D=1$. Embedding additional variables allows the evolution of generative trajectories controlled by scalar norms, offering a novel trade-off mechanism between robustness and rigidity. Specifically, larger $D$ values correspond to more concentrated distributions around the data, improving rigidity, while smaller $D$ values enhance robustness but introduce challenges in learning from heavy-tailed distributions.

Objective Function and Training Methodology

The framework dispenses with large batch requirements from traditional PFGM by introducing a new perturbation-based objective. This method remains unbiased, efficient, and permits conditional generation with paired training samples. The model leverages isotropic perturbation kernels, effectively aligning with denoising score matching strategies used in diffusion models, thereby providing a seamless transition between them in the $D{\to}\infty$ limit.

Empirical Evaluation and Implications

PFGM++ demonstrates empirical superiority over state-of-the-art models on benchmark datasets like CIFAR-10 and FFHQ $64{\times}64$, achieving FID scores of 1.91 and 2.43 respectively, under certain configurations of $D$. Notably, it outperforms when $D=2048$ in class-conditional settings. The adaptability of $D$ allows the model to balance robustness and learning rigour effectively, outperforming both its predecessor frameworks by optimizing this parameter.

The paper also shows that PFGM++ models with smaller $D$ exhibit greater resilience to errors associated with noise, sampling step size, and post-training quantization, making them potentially more applicable in constrained computational environments.

Theoretical and Practical Implications

The introduction of PFGM++ could influence broader developments in generative modeling. The key insight lies in its flexibility to interpolate between known model types, preserving the strengths of diffusion models’ stability and PFGM’s robustness. This flexibility implies a potential for extending model utility across various domains, possibly enhancing capabilities in high-dimensional data synthesis and complex multimodal distributions.

Future Prospects

The findings open several avenues for future research, such as extending this framework to dynamically adapt $D$ during training to balance robustness and efficiency optimally across different applications. Another future consideration could be investigating stochastic adaptations of the framework, leveraging the inherent randomness for improved sample diversity in domains like audio and biological data generation.

Overall, PFGM++ represents a significant advance in the unified design of generative models, exhibiting potential to impact both theoretical understanding and practical applications of machine learning in numerous fields.