糖心原创

Well-posedness and stability of physics-informed data fitting

 

This is a project in analysis, with numerical experiments to keep it honest. Suppose you have a few measurements and a PDE you mostly trust but whose sources or boundary data you do not know exactly. A popular idea, behind physics-informed neural networks, is to fit a model that matches the data and satisfies the equation at the same time, by adding the two errors and minimising. It is least squares with an extra term. Sometimes this works very well and sometimes it gives nonsense. Adding data can rescue an equation that is ill-posed on its own, and adding the equation can stop a fit from chasing noise, but only if the two are measured in the right norms and balanced well. You will make this precise: when the combined problem has a unique solution that depends continuously on the data, how errors propagate, and what this means for physics-informed networks and for classical numerical methods. Along the way you will learn the functional analysis behind modern PDE theory.