让神经网络权重随任务变化，在低维流形上平滑调节，提升泛化能力。

One frequently wishes to learn a range of similar tasks as efficiently as possible, re-using knowledge across tasks. In artificial neural networks, this is typically accomplished by conditioning a network upon task context by injecting context as input. Brains have a different strategy: the parameters themselves are modulated as a function of various neuromodulators such as serotonin. Here, we take inspiration from neuromodulation and propose to learn weights which are smoothly parameterized functions of task context variables. Rather than optimize a weight vector, i.e. a single point in weight space, we optimize a smooth manifold in weight space with a predefined topology. To accomplish this, we derive a formal treatment of optimization of manifolds as the minimization of a loss functional subject to a constraint on volumetric movement, analogous to gradient descent. During inference, conditioning selects a single point on this manifold which serves as the effective weight matrix for a particular sub-task. This strategy for conditioning has two main advantages. First, the topology of the manifold (whether a line, circle, or torus) is a convenient lever for inductive biases about the relationship between tasks. Second, learning in one state smoothly affects the entire manifold, encouraging generalization across states. To verify this, we train manifolds with several topologies, including straight lines in weight space (for conditioning on e.g. noise level in input data) and ellipses (for rotated images). Despite their simplicity, these parameterizations outperform conditioning identical networks by input concatenation and better generalize to out-of-distribution samples. These results suggest that modulating weights over low-dimensional manifolds offers a principled and effective alternative to traditional conditioning.