Zhongyan Ouyang, Weixin Liao, Mingquan Feng, Yehui Tang, Junchi Yan
Neural subspace solvers offer a promising way to accelerate the large sparse linear systems arising from PDE discretizations, but their value depends on generating effective correction directions at low computational overheads. We study this problem from a residual-image viewpoint: a correction subspace is useful when its image under the system operator captures the current residual. Guided by this view, we propose Gate-RINS, which generates multiple correction directions from shared polynomial residual probes and modulates them with a lightweight residual- and coordinate-dependent pointwise gate. We also introduce a hybrid schedule that uses GRANS-style graph controllers for the first correction block and Gate-RINS thereafter, combining an expressive graph based initial step with a lower-cost corrector as the solve progress. Theoretical analysis establishes a unified residual-image and dynamic approximate-inverse perspective for Gate-RINS and G-RANS, showing that a single neural-generated direction can emulate the residual quality of fixed-step GMRES. Numerical experiments on six PDE benchmarks demonstrate that Gate-RINS reaches a relative residual of $10^{-5}$ with 1.6-2.3$\times$ and 2.2-3.0$\times$ less wall-clock time than G-RANS and GMRES respectively. For Helmholtz systems with meshsize up to $10^5$, Gate-RINS also achieves the lowest residual and outperforms a GPU-accelerated parallel GMRES, while GRANS fails to reach relative residual of $10^{-4}$ in the same time budget.
Financial agents must form forecasts and investment judgments from decision-time evidence, with predictions evaluated against subsequent outcomes. Existing benchmarks assess financial knowledge, research workflows, or forecasting, but evaluating the full research process requires a strictly frozen information set and professional evaluation criteria. We introduce Investor's Last Exam (ILE), an equity-research benchmark led by experienced buy-side investors and fund managers. Its three components are a versioned financial database spanning 3.43 million document versions and nine source categories, supplemented by structured market data, with validated timestamps governing retrieval and computation; a task taxonomy organized around professional analyst deliverables; and an expert-led framework linking scoring criteria to sources, calculations, and assumptions through financial dependency graphs. Evaluation spans 86 assignments and jointly assesses research quality, forecast accuracy, and simulated investment performance. GPT-6 Astra and Claude Opus 5.5 lead overall at 68.1 and 66.9 out of 100, yet even Astra scores only 52.5 on Forecast & Model. Our results show that even state-of-the-art models fail to translate overall research capability into predictive performance. ILE shifts the evaluation unit from isolated capabilities to the evidence--assumption--judgment chain, allowing researchers to locate where analytical advantages fail to translate into forecasts and investment judgments, and providing a traceable, diagnostic foundation for training and evaluating financial agents.