GAIA 能用一个模型同时搞定正反问题，在七种几何任务上都拿了第一，尤其反问题精度提升明显，做 PDE 模拟的可以看看。
GAIA 是一种几何自适应积分自编码器，用于任意几何域上的偏微分方程正问题和反问题的算子学习。它通过将几何边界和内部场分布编码为几何令牌，并利用交叉注意力机制使积分变换核局部适应几何特征，无需重新训练或迭代优化。在 7 个 2D 和 3D 基准测试中，包括电机阻抗断层扫描（EIT）、光学断层扫描、变化几何上的 3D Darct 流以及机械零件泊松 BVP 基准（MCB），GAIA 在所有反问题和 BVP 任务上取得 SOTA。与次优方法相比，在机翼流重建任务中降低了 64% 的中位相对 L2 误差，在 EIT 上降低了 27%，且在 MCB 每个形状类别上超越所有基线。
GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems
Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.