DSEM — Differentiable Spectral Element Software

PyTorch-based differentiable spectral element method (SEM) for PDE-constrained inverse and optimization problems.

This repository contains the exact code and data used for the manuscript

Differentiable spectral element software for PDE-constrained inverse problems: automatic adjoint consistency and reduced Hessian conditioning Fuchang Wang, Huirong Cao. Computer Physics Communications (submitted).

What it does

A unified, high-order spectral element library (code/sem_lib.py) that supports:

• 1D / 2D / 3D tensor-product GLL spectral elements (diagonal GLL mass matrix, strong Dirichlet conditions);
• forward solves, reverse-mode automatic differentiation (PyTorch), and hand-coded discrete adjoints, with AD–adjoint consistency verified to machine precision;
• dense, sparse, and element-wise matrix-free solution modes;
• PDE-constrained optimization (L-BFGS) with Tikhonov regularization, including a reduced-Hessian conditioning analysis;
• representative nonlinear steady residuals (steady Burgers, 2D nonlinear diffusion).

It is intended as a reusable differentiable high-order PDE optimization architecture: for a new residual, one supplies the element residual and the framework obtains gradients automatically, without deriving or maintaining a problem-specific continuous or discrete adjoint.

Layout

Path	Contents
code/	Final unified-library scripts (reproduce every table and figure)
results/	Machine-readable JSON outputs used by the manuscript
figures/	Figure scripts and the vector figures (PDF)
oms_draft/	Earlier draft scripts (consistent-mass formulation, OMS-era) retained for provenance only — not used in the CPC manuscript
CITATION.cff	Citation metadata (Zenodo DOI will be added at release)
environment.yml	Conda environment (Python 3.11, PyTorch ≥ 2.1, NumPy 2.x)

Reproduction

conda env create -f environment.yml
conda activate dsem-cpc
cd code
python verify_all.py              # retained verification tables
python verify_lshape_opt.py       # L-shaped-domain optimization
python verify_hessian_cond.py     # reduced-Hessian conditioning (Table)
python calibrate_B_scaling.py     # mu_max(B) ~ p^4 h^-2 calibration (Thm 2)
python expA_2d_optimization_benchmark.py   # AD vs hand adjoint vs FD
python expB_matrixfree_scaling.py          # matrix-free scaling, log-log fits
python expC_2d_burgers.py                  # 2D steady nonlinear residual
cd ../figures && python make_figures.py    # all figures from results/*.json

All arithmetic is float64 (CPU). Timings in the paper were measured on an Intel Core i7-8850H, 32 GB RAM, single process, and are reported for that environment.

Archiving

The exact commit behind the manuscript is tagged as Release v1.0.0 and archived with a persistent DOI:

MIT license.

Citation

See CITATION.cff (BibTeX-friendly fields). When the Zenodo DOI is minted, cite the Zenodo record; until then, cite the GitHub repository.