统一物理反向传播
Unifying Physical Backpropagation
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中文总结 AI 辅助
本文基于伴随方法构建统一理论，明确物理系统可计算自身性能梯度的充分条件，复现多种物理学习算法，为物理学习算法提供统一理论基础。
中文摘要 AI 辅助
物理计算系统利用器件动力学进行计算，但其基于梯度的优化颇具挑战：通过数字孪生进行反向传播存在模型-现实差距。设备端梯度计算可解决该问题，已有少量理论与实验研究提出实现方法，但始终缺乏一种统一理论，以明确物理系统何时能计算自身性能的梯度。本文基于伴随方法构建了这样的统一理论：我们确定了充分条件，即执行计算的同一硬件可生成形式上精确梯度所需的伴随场。线性系统与非线性系统遵循截然不同的条件：对于线性系统，只要保持互易性，阻尼或增益是可允许的；对于非线性轨迹系统，充分条件为线性化系统的互易性以及时间反转镜的存在。算法层面，非线性情形需要无穷小的扰动，而线性系统可采用有限振幅实验。我们复现了平衡传播、哈密顿回声反向传播、全前向模式训练以及集成光子与自由空间光学系统中的原位梯度方法。我们进一步表明，互易性仅是更广义交织条件的最简单实例，该条件将精确设备端梯度计算扩展至一类非厄米、非互易系统；进一步的广义化还包括时变参数、昂萨格互易动力学以及非线性PT对称薛定谔方程。本研究为形式上精确的物理学习算法提供了统一理论基础，也为在各类物理系统中构建此类算法提供了模板。
英文摘要
Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from a model-reality gap. On-device gradient computation could resolve this issue, and a handful of theoretical and experimental studies have proposed ways to achieve it. Yet a unifying theory identifying when a physical system can compute the gradient of its own performance has been missing. Here we develop such a unification based on the adjoint method: we identify sufficient conditions under which the adjoint field required for formally exact gradients can be generated on the same hardware that performs the computation. Linear and nonlinear systems obey fundamentally different conditions: for linear systems, damping or gain is admissible provided reciprocity is preserved. For nonlinear trajectory systems, the sufficient conditions are reciprocity of the linearized system and the existence of a time-reversal mirror. Algorithmically, the nonlinear case requires infinitesimal nudging, whereas linear systems admit a finite-amplitude experiment. We recover (quantum) Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. Finally, we show that reciprocity is a special case of more general intertwining conditions. For linear systems, these permit exact on-device gradients in a class of non-Hermitian, non-reciprocal systems. For nonlinear trajectories, they combine with generalized time-reversal mirrors to cover, e.g., PT-symmetric equations. The framework also includes time-dependent parameters and Onsager-reciprocal dynamics, providing a unified basis for formally exact physical learning.
发表机构
- Princeton University（普林斯顿大学）
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