One of my recent projects was showing how to find solutions of PDEs on complex geometries without needing a mesh by fitting equation residuals to a spectral basis (Roggeveen & Brenner, 2025). Here, I show some animated versions of the time-dependent solutions presented in the article.
Wave equation
We solve the wave equation on a peanut shaped geometry and compare it to a solution generated by COMSOL. We use 13 Chebyshev polynomials in each of $x$, $y$, and $t$.
Diffusion on a sphere
We solve the diffusion equation confined to the surface of a sphere by embedding the spherical surface into a 3D space and fitting 11 Chebyshev polynomials in $x$, $y$, $z$, and $t$.
Forced heat equation
We solve the heat equation on a disc while simulatneous solving for a parameterization of the boundary condition that causes the intial heat distribution to reflect about the line $x=0$.
Optimized transport
We solve the advection-diffusion equation for some scalar field $c$ while finding a solenoidal flow field $\mathbf{u}$ that forces the concentration field to spell out Harvard.
References
2025
Meshless solutions of PDE inverse problems on irregular geometries
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral bases on arbitrary spatiotemporal domains, whereby the basis is defined on a hyperrectangle containing the true domain. We find the coefficients of the basis expansion by solving an optimization problem whereby both the equations, the boundary conditions and any optimization targets are enforced by a loss function, building on a key idea from Physics-Informed Neural Networks (PINNs). Since the representation of the function natively has exponential convergence, so does the solution of the optimization problem, as long as it can be solved efficiently. We find empirically that the optimization protocols developed for machine learning find solutions with exponential convergence on a wide range of equations. The method naturally allows for the incorporation of data assimilation by including additional terms in the loss function, and for the efficient solution of optimization problems over the PDE solutions.