PDE solver | James V. Roggeveen

This project develops a meshless method for solving PDE inverse and optimization problems on complex geometries by fitting equation residuals to a spectral basis (Roggeveen & Brenner, 2025). The examples below show time-dependent solutions and inverse-design problems from the current draft.

I am currently revising and extending this work, with an emphasis on using the solver as a practical engine for scientific inverse problems and optimization workflows. A useful pattern has been to require coding agents to pass independent numerical checks before reporting results, which helps catch invalid implementations while extending the solver to new applications.

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

James V. Roggeveen , and Michael P. Brenner

Oct 2025

Abs arXiv

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.