# Neural Operators for PDEs

$$\mathcal{N}^{\theta}: \mathcal{A}(\Omega_{\alpha};\mathbb{R}^{d_a}) \rightarrow \mathcal{U}(\Omega_{\alpha};\mathbb{R}^{d_u})$$

where $\theta \in \Theta$ denotes the neural operator’s parameters, $\Omega_{\alpha} \subset \mathbb{R}^d$ (or a $d$-dimensional manifold $\mathcal{M}^d$) represents the spatial domain on which the functions are defined and $\alpha \in \mathscr{A}$ denotes the shape of the domain. The dimensions $d_a$ and $d_u$ denote the respective sizes of the input and output function spaces, often subspaces of Sobolev or Banach spaces.

Neural operators can be applied to various problems such as those described by continuous functions $C(\Omega;\mathbb{R}^{d_a})$ or Sobolev spaces $H^s(\Omega;\mathbb{R}^{d_a})$ for some $s \ge 0$. The neural operator $\mathcal{N}^{\theta}$ is an approximation of a true target operator $\mathcal{N}$, e.g., the solution operator of a partial differential equation (PDE) obtained by training on input-output function pairs $(a_i, u_i)_{i=1}^{m}$, where $a_i \in \mathcal{A}$ and $u_i = \mathcal{N}(a_i) \in \mathcal{U}$. These pairs could be simulation data representing a known, high-fidelity numerical approximation of the PDE.

My research on this topic aims at developing robust computational and mathematical frameworks for operator/PDE learning on arbitrary domains and enforcing known symmetries of the problem to enhance the data efficiency of these ideas.

Figure 1: Neural operator prediction of solution to the anisotropic 2D heat equation compared with numerical solution.

Figure 2: Neural operator vs. numerical solver for reaction-diffusion on anisotropic 2D rectangle.

Figure 3: Cardiac Electrophysiology Simulation using Neural Operators (left: ground truth, right: prediction).