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SUMMARY:From Physics-informed Neural Networks to Numerically Verified Proo
 fs in Geometric Analysis - Daniel Platt (Imperial College London)
DTSTART:20260421T130000Z
DTEND:20260421T131500Z
UID:TALK246646@talks.cam.ac.uk
DESCRIPTION:There are many problems in geometric analysis where one wants 
 to prove existence of a solution of an elliptic PDE on a compact manifold.
  The curvature of the geometry makes it difficult to apply classical PDE s
 olvers and recently Physics-informed Neural Networks (PINNs) have been app
 lied with some success to study many equations through numerical simulatio
 ns\, and I will list some of them. It would then be desriable to turn thes
 e simulations into a numerically verified proof. The equations are often m
 uch better behaved compared to some analysis problems on Euclidean domains
  where numerically verified proofs were carried out. But the curvature ent
 ers in a difficult way into linear estimates\, which makes it difficult to
  run such proofs on closed manifolds\, even if the equations are simple. I
  will compare a recent numerically verified proof on the sphere with ongoi
 ng work on a more complicated manifold and highlight where the challenges 
 lie.
LOCATION:Seminar Room 1\, Newton Institute
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