这个新方法LiL-Q用凸优化替代了PINNs的非凸训练，收敛快、参数少，在Navier-Stokes等难题上效果拔群，值得关注。
该论文提出LiL-Q方法，通过Bellman-Kalaba拟线性化将非线性PDE转化为一系列线性子问题，每个子问题用线性可学习表示（LiL）离散化并直接通过QR分解求解。LiL表示包括随机特征极限学习机、谱多项式基和三角展开，实现为物理信息神经网络（PINNs）。在7个基准测试（Bratu、粘性Burgers、Buckley-Leverett、平面应变弹性、2D/3D不可压缩Navier-Stokes、异质渗透率Darcy流）上，LiL-Q通常在个位数外部迭代内收敛，参数数量比现有PINN求解器少两个数量级时仍能达到或超越其精度。当精确解在试验空间内时，单次求解即可恢复至机器精度。
A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks
We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.