DEFlow: Deep Equilibrium Flow Maximum-Entropy Reinforcement Learning

Diffusion models have recently emerged as expressive policy representations for online reinforcement learning (RL). However, their iterative generative processes introduce substantial training and inference overhead. To overcome this limitation, we propose to represent policies using MeanFlow models, a class of few-step flow-based generative models, to improve training and inference efficiency over diffusion-based RL approaches. To promote exploration, we optimize MeanFlow policies under the maximum entropy RL framework via soft policy iteration, and address two key challenges specific to MeanFlow policies: action likelihood evaluation and soft policy improvement. Experiments on MuJoCo, DeepMind Control Suite and HumanoidBench benchmarks demonstrate that our method, Mean Flow Policy Optimization (MFPO), achieves performance comparable to or exceeding current diffusion-based baselines while considerably reducing training and inference time. Our code is available at https://github.com/dongxiaoyi-xyz/MFPO.

Diffusion models and Flow Matching can represent flexible distributions over robot actions, but many reinforcement learning algorithms also require the probability of each action. We introduce PrT-Flow (Probability-Tractable Flow Matching), which computes probability density while retaining multimodal, multi-step action generation. Its Jacobian-Tractable Transformer (JTT) makes each sampling step invertible and enables efficient computation of its Jacobian determinant. We train these finite steps with Conditional Euler-step Training (CET), which supervises their displacement over one step during sampling rather than their instantaneous velocity. We prove that, under certain assumptions, CET has the same minimizers as fitting the Euler-solver updates of Flow Matching. PrT-Flow achieves 96% success on Robomimic Lift after 300 epochs of imitation learning, matching the diffusion policy reference, and 96% success on OGBench Cube-Triple after 600k online steps. These results demonstrate that invertible networks with tractable probability densities can learn effective robot policies and support SAC-style actor-critic training. Our code is available at https://github.com/Sunlight-zero/PrT-Flow .

Real-world reinforcement learning for robotic manipulation remains challenging, and this difficulty is amplified for flow matching policies: policy gradients must be backpropagated through time (BPTT) along the multi-step ODE that maps noise to actions, which is computationally prohibitive and numerically fragile. We propose FlowDPG, a DDPG-style method for flow matching policies that bypasses BPTT entirely. FlowDPG evaluates the critic gradient at a one-step estimate of the clean action and distills the resulting correction into the velocity field at training time, leaving multi-step inference unchanged. Intuitively, it combines two complementary vectors: the demonstration-driven velocity that keeps the action feasible, and the critic-driven correction that steers it toward higher value. Our contributions are threefold: (1) a BPTT-free distillation framework for stable DDPG-style improvement of flow matching policies, (2) a formal connection between the FlowDPG update and the vanilla deterministic policy gradient via three explicit approximations, and (3) real-world validation on two long-horizon tasks on different robots. FlowDPG reaches 92% end-to-end success on dual-arm AirPods assembly with Franka arms and an 86% rubric score on scrambled-egg cooking with YAM arms, substantially outperforming recent RL methods. Videos and more results: https://flowdpg.github.io.