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Explore cutting-edge mathematical research combining modern computational methods with classical analysis to understand complex solutions of partial differential equations (PDEs). Learn how machine learning techniques, specifically physics-informed neural networks (PINNs), are revolutionizing the discovery of new solutions to nonlinear PDEs with unprecedented accuracy. Discover the crucial interplay between computational and analytical methods that enables researchers to uncover the complete spectrum of PDE solutions, with particular emphasis on unstable solutions that traditional methods struggle to identify. Examine how computer-assisted proof techniques can transform numerical discoveries into rigorous mathematical theorems, bridging the gap between computational exploration and formal mathematical verification. While the presentation focuses on fluid mechanics equations including Euler and Navier-Stokes equations, understand how these innovative methodologies extend to broader classes of PDEs across various mathematical and physical applications.
