Motivation
Independent components in ML
1.
Neural Architectures
1.1.
Multilayer perceptron
1.2.
Empirical low-rank structure
2.
Loss Functions
2.1.
Residual loss
2.2.
Regression loss
2.3.
Variational loss
3.
Optimization Methods
3.1.
Gradient descent
3.2.
Ensemble Kalman Filter
3.2.1.
Minimizing Rosenbrock
3.2.2.
Minimizing ODE PINN loss
Extension to Variational loss
4.
Deep Ritz Method
4.1.
Ritz-Rayleigh quotient and Laplace eigenfunction
4.1.1.
Triangle
4.1.2.
Square
4.1.3.
Circle
4.1.4.
Snowflake
4.2.
Boundary residual loss
4.2.1.
Triangle Failed
4.3.
Obstacle Problems
Inverse Problems in PDE
5.
Inverse Problems in PDE
5.1.
Unknown Parameters in PDE
5.1.1.
"Heat conductivity" from temperature
6.
Future work
Light
Rust
Coal
Navy
Ayu
Deep Learning Methods for Computational Math
Triangle (Failed due to boundary)