Energy Waveify and Redistribution for Test-Time Adaptation: A Control System Perspective

This work tackles a key challenge in test-time energy adaptation: prohibitive time overhead arising from recent stateof-the-art test-time adaptation (TTA) methods, which are built on energy models relying on iterative Monte Carlo or Langevin dynamics sampling with multiple stochastic updates per test instance to approximate energy gradients. We tackle the problem from an innovative controlsystem perspective by i) describing the energy as a complexvalued wave, where the amplitude encodes energy uncertainty and the phase characterizes its evolution, and ii) maintaining a time-dependent wave equation that interprets TTA as a control-system evolution process. By enforcing the control-system law of probability current conservation, our method directs probability current away from highenergy (error-prone) regions toward low-energy (accurate) ones, achieving adaptive energy redistribution without additional stochastic sampling while preserving the overall normalization of the energy landscape. Experimentally, the proposed method significantly outperforms baseline methods across several public benchmark datasets, with adaptive time being only 1/3 ∼ 1/7 of that required by the compared Top-1 to Top-3 baselines. Code is available at https://github.com/wongzbb/APT.