Control Theory

Classical data-driven methods in machine learning tend to be data-inefficient and they discard expert modeling knowledge. However, during the last decade a different approach has emerged under the name physics-informed machine learning. First posted in 2017 and published in 2019, physics-informed neural networks (PINNs) [6] combined deep learning with prior knowledge of the governing equations by directly learning the solution using the PDE residual as a soft penalty in the loss. In 2018, Neural ODEs [1] took a complementary route of embedding the physics in the model itself: the right-hand side of an ordinary differential equation $\dot y=f(y,t)$ is replaced by a neural network $f\approx f_\theta$, trained to reproduce observed data. The same principle was soon extended to stochastic differential equations [7] and to partial differential equations [5, 11, 12] — the latter is what we call here a Neural PDE. ...

Machine learning feels recent, but one of its core mathematical ideas dates back to 1952, when Richard Bellman published a seminal paper titled “On the Theory of Dynamic Programming” [6, 7], laying the foundation for optimal control and what we now call reinforcement learning. Later in the 50s, Bellman extended his work to continuous-time systems, turning the optimal condition into a PDE. What he later found was that this was identical to a result in physics published a century before (1840s), known as the Hamilton-Jacobi equation. ...

% History: Linear Quadratic Regulator problem 1960s Consider the following dynamical system $$ \begin{align*} \dot x &= Ax + Bu, \qquad x(0)=x_0\\ y &= Cx, \qquad 0\leq t\leq T \end{align*} $$where $u\in L^2(0,T; U)$ and $y\in L^2(0,T; Y)$ and $U$ and $Y$ are Hilbert spaces. % Hypothesis, well-posedness % Differential Riccati equation Examples Parabolic Consider the equation $$ \dot x = Ax + Bu, \qquad y = Cx $$ where $A$ is self-adjoint on a real Hilbert space. We assume that $A$ has a compact resolvent operator and that the spectrum of $A$ consists of a strictly decreasing sequence $\lambda_n, n\in \mathbb{N}$ of real eigenvalues with associated eigenvector $\phi_n\in H$ with $\|\phi_n\|=1$. ...

I think there are 3 major milestones remaining for humanity, and one of them is clean and abundant energy. Fusion energy has the potential to provide a nearly limitless source of clean energy by replicating the processes that power the sun. However, achieving controlled fusion reactions on Earth has proven to be a formidable and very challenging task. The most well-known fusion prototype is the tokamak, which uses a magnetic field to confine the plasma within a toroidal chamber (doughnut-shaped). ...