A central challenge in game theory and learning systems such as GANs is understanding which algorithms can efficiently compute equilibria across the heterogeneous landscape of games. Equilibrium computation is typically studied solver by solver and game class by game class, yielding strong local guarantees but a fragmented view of solver behaviour. Existing discrete taxonomies often provide an incomplete account of where algorithms succeed. We study this problem through a solver-game map linking games to effective solver dynamics. Classical theory identifies isolated regions of this map but provides limited insight into intermediate or overlapping regimes, suggesting that solvability is governed by latent structural properties defining a continuous solver-aligned geometry of games. We formalise this perspective through structure-aware solver synthesis. A learned structure recogniser maps each game to a low-dimensional solver-aligned representation, and a policy maps this representation to effective primitive mechanisms, adapting solver behaviour across regimes. This reveals regions where particular solver dynamics are effective and where mixtures of primitives are required rather than a single dominant solver. A bounded residual acts as a local corrector and diagnostic signal for incomplete solver bases or representations. The framework yields both an adaptive solver and an analytical lens: games with similar optimisation dynamics cluster together, revealing continuous regions of algorithmic validity and overlapping solver behaviour. Empirically, we show that fixed primitives exhibit systematic regime mismatch, while the learned representation organises game space into a structured cartography aligned with solver behaviour. These results suggest viewing equilibrium computation as the joint problem of learning solver mechanisms and mapping the geometry of solvability.

翻译：博弈论与生成对抗网络等学习系统中的核心挑战在于：理解哪些算法能在异质化的博弈格局中高效计算均衡解。均衡计算通常按求解器类别与博弈类型逐一分析，虽能给出强局部保证，却导致求解器行为的碎片化认知。现有的离散分类体系往往无法完整描述算法的适用场景。我们通过构建求解器-博弈映射图来研究该问题，该图将博弈与有效求解器动力学相关联。经典理论虽能识别该映射图上的孤立区域，但对中间或重叠机制域的认知有限，这表明可解性受潜在结构属性支配，这些属性定义了与求解器对齐的连续博弈几何结构。我们通过结构感知求解器合成将这一观点形式化：学习型结构识别器将每个博弈映射至低维的求解器对齐表示，策略网络将该表示映射为有效基元机制，从而自适应调整不同机制域中的求解器行为。该方法揭示了特定求解器动力学有效的区域，以及需要基元混合而非单一主导求解器的场景。有界残差作为局部校正器与诊断信号，用于检测不完整的求解器基组或表示。该框架同时具备自适应求解器与分析透镜双重功能：具有相似优化动力学的博弈会聚集呈现，揭示连续算法有效性区域与重叠求解器行为。实验表明，固定基元存在系统性机制错配，而学习表示可将博弈空间组织成与求解器行为对齐的结构化图谱。这些结果表明，应将均衡计算视为求解器机制学习与可解性几何结构映射的联合问题。