让自动编码器在不同条件下保持几何一致性，稳定捕捉物理系统的低维结构。
Context-dependent manifold learning: A neuromodulated constrained autoencoder approach
- 用上下文驱动的超网络调节编码器激活参数，保持投影恒等性。
- 在16自由度摆和Lorenz96系统上，重建精度和几何一致性均优于六种基线方法。
- 特别适合需要迭代应用、跨物理参数变化的系统建模任务。
许多物理系统具有随外部参数变化的低维结构：如机器人的连杆长度、流体的受力常数或流动中的雷诺数改变底层流形，但其内在维度保持不变。约束自编码器（cAE）通过幂等编码-解码映射学习此类流形，这一性质是无约束自编码器无法具备的，尤其在模型需迭代使用时至关重要。然而，现有使cAE具备上下文依赖性的方法——如将上下文拼接到输入或仿射调制隐藏层激活——会破坏编码-解码的幂等性，恰恰在最需要该性质的场景中牺牲了投影保证。为此，我们提出神经调制约束自编码器（NcAE），通过上下文驱动的超网络调节cAE的激活斜率与偏置，恢复在上下文变化下的幂等性。本文证明：对任意上下文（包括训练时未见的），重构映射仍为幂等投影，所学流形拓扑保持不变，且上下文扰动导致流形平滑变化。我们在一个具有上下文依赖耦合的16自由度摆和经历分岔的Lorenz96系统上验证该方法。NcAE在重建质量、幂等性和潜在几何度量上达到或超越六种基线，且唯一能从结构上保持几何一致性。NcAE因此提供了一种在物理参数族间稳定的、保持几何特性的坐标系。
原文摘要 · Abstract (English)
Many physical systems exhibit a low-dimensional structure that varies with external parameters: link lengths in a robot, forcing constants in a fluid, or Reynolds numbers in a flow shift the underlying manifold while preserving its intrinsic dimension. Constrained AutoEncoders (cAEs) learn such manifolds through an idempotent encoder-decoder projection, a property that unconstrained autoencoders cannot match and that is essential whenever the model is applied iteratively. However, the standard strategies for making a cAE context-dependent, namely concatenating the context to the input or affinely modulating hidden activations, break the encoder-decoder idempotency, sacrificing the projection guarantee precisely in the setting where it would be most valuable. To restore this guarantee under context variation, we developed the Neuromodulated Constrained Autoencoder (NcAE), which modulates the activation slope and bias of a cAE through a context-driven hyper-network. This paper presents the NcAE, its theoretical foundation, and its empirical validation. We prove that for every context, including contexts unseen at training time, the reconstruction map remains an idempotent projection, the topology of the learned manifold is invariant, and context perturbations induce smooth changes in the manifold. We evaluated our approach on a 16-DoF pendulum with context-dependent coupling and the Lorenz96 system across a bifurcation. The NcAE matched or exceeded the best of six baselines on reconstruction, idempotency, and latent-geometry metrics, while being the only architecture that preserves geometric consistency by construction. The NcAE thereby provides a stable, geometry-preserving coordinate system across families of physical regimes.
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