Search papers, labs, and topics across Lattice.
This paper introduces the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel framework that eliminates the need for backpropagation in solving nonlinear differential equations. By analytically evaluating local Jacobians and reformulating the physical loss into a linearized algebraic subspace, PI-SCM enables the direct computation of optimal network weights using generalized linear least squares solvers. Experimental results show that PI-SCM not only accelerates training significantly compared to traditional Physics-Informed Neural Networks (PINNs) but also maintains high predictive accuracy and robust parameter identification.
Training neural networks for nonlinear differential equations just got exponentially faster鈥攚ithout backpropagation.
While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Machine (PI-SCM), a novel backpropagation-free framework for both forward and inverse problems in differential equations. The core mathematical contribution lies in the analytical evaluation of local Jacobians for nonlinear differential operators, which facilitates a linearized representation of the physical loss and projects it into a unified, linearized algebraic subspace. This reformulation allows for the explicit determination of optimal network weights via a sequence of generalized linear least squares solvers, effectively bypassing the iterative traps of traditional nonlinear optimizers. We develop a progressive algorithmic suite comprising localized construction (PI-SC-I), sliding-window updating (PI-SC-II), and global updating (PI-SC-III), and rigorously establish their universal approximation properties. Extensive experiments demonstrate that PI-SCM achieves high-fidelity predictive accuracy and robust parameter identification while accelerating the training process by orders of magnitude compared to standard PINNs. Our work provides a highly efficient and scalable foundation for next-generation, real-time Scientific Machine Learning applications.
