A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks
arXiv:2606.18175v1 Announce Type: cross Abstract: 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, s
The proliferation of advanced AI techniques and the demand for more efficient solutions to complex scientific and engineering problems drive the development of novel PDE solvers.
This method offers a potentially significant advancement in solving nonlinear partial differential equations, a bottleneck in many scientific and engineering fields, by leveraging the strengths of both traditional numerical methods and physics-informed neural networks.
The ability to accurately and efficiently solve complex nonlinear PDEs could accelerate research and development in areas from fluid dynamics to materials science, reducing computational costs and time to insight.
- · AI researchers
- · Computational scientists
- · Engineering sectors
- · Supercomputing facilities
- · Traditional numerical methods reliant on manual discretization
- · Sectors heavily invested in legacy PDE solvers
More accurate and faster simulations across various scientific and engineering disciplines become possible.
This could lead to breakthroughs in areas requiring high-fidelity modeling, such as drug discovery or climate modeling.
The reduced computational burden might lower barriers to entry for complex simulations, democratizing advanced research and development.
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Read at arXiv cs.LG