AI 中文总结
针对神经预测器原像难以表示和优化的问题，提出TRIO框架，通过径向变换分解实现表达性前向建模、精确原像恢复和可处理全局优化的统一。
AI 中文摘要
现代神经预测器能够建模高度非线性的映射，但许多科学和工程任务需要在相反方向上进行推理：给定性能或安全水平，目标是刻画原像，即满足期望目标水平的完整输入集合，并优化该集合。然而，对于表达力强的神经预测器，这类原像通常没有显式表示，且恢复或优化成本高昂。这在前向预测的表达力、原像逼近的准确性以及下游任务中对原像的可处理优化之间构成了一个根本性的三方挑战。我们提出了TRIO（原像学习和逆优化的可处理表示），这是一个学习表示的框架，通过构造使这些目标兼容。我们的关键贡献是原像分解：前向模型通过非线性径向变换（包括神经网络）保持表达力，而在求逆时，每个变换简化为单个标量半径，从而产生简单的几何水平集。这产生了一个显式的几何表示，可重复用于对原像的下游优化，并且对于线性目标，我们证明了这允许闭式全局解。最后，我们证明了一个通用逼近定理，表明TRIO可以任意逼近任何连续前向映射及其整个可能不连通、非凸的原像族。因此，TRIO通过设计结合了表达性前向建模、精确原像恢复和可处理的全局下游原像优化。
英文摘要
Modern neural predictors can model highly nonlinear maps, but many scientific and engineering tasks require reasoning in the opposite direction: given a performance or safety level, the goal is to characterize the preimage, that is, the complete set of inputs which meet the desired target level and optimize over that set. For expressive neural predictors, however, such preimages typically have no explicit representation and are expensive to recover or optimize over. This creates a fundamental three-way challenge between expressive forward prediction, accurate preimage approximation, and tractable optimization over the preimage for downstream tasks. We introduce TRIO (tractable representations for preimage learning and inverse optimization), a framework for learning representations that make these objectives compatible by construction. Our key contribution is a preimage factorization: the forward model remains expressive through nonlinear radial transformations (including neural networks), while, under inversion, each transformation reduces to a single scalar radius, which yields simple geometric level sets. This yields an explicit geometric representation that is reusable for downstream optimization over the preimage, and, for linear objectives, we show that this admits a closed-form global solution. We finally prove a universal approximation theorem which shows that TRIO can approximate any continuous forward map and its entire family of potentially disconnected, nonconvex preimages arbitrarily well. Hence, TRIO combines expressive forward modeling, exact preimage recovery, and tractable global downstream optimization over preimages by design.
