# PDE Solving

High-accuracy gravitational-wave source modeling has long avoided machine learning because neural networks carry a noise floor. Researchers trained a graph neural network as a preconditioner for an iterative solver, leaving the traditional GMRES to enforce final accuracy while the network only accelerates convergence. On the high-frequency elliptic equations for asymmetric-mass black hole binaries, the hardest parameter regions need 2 to 15 fewer GMRES iterations than basic ILU and 2 to 4 times less wall time than the next best method.

High-accuracy gravitational-wave source modeling has long avoided machine learning because neural networks carry a noise floor. Researchers trained a graph neural network as a preconditioner for an iterative solver, leaving the traditional GMRES to enforce final accuracy while the network only accelerates convergence. On the high-frequency elliptic equations for asymmetric-mass black hole binaries, the hardest parameter regions need 2 to 15 fewer GMRES iterations than basic ILU and 2 to 4 times less wall time than the next best method.

Recovering a partial differential equation from sampled solution data demands expert work in equation discovery, coefficient estimation and numerical solver configuration. RSI-PDEbench turns that workflow into a benchmark of 1,001 cases over 42 equation templates, split into noise-free, 50% Gaussian-noise and unseen-template-with-shifted-coefficient settings; the accompanying RSI-PDE agent shortlists candidate equations by residual, refines coefficients and solvers with relative feedback, and keeps improvements behind a validation gate. Across four frontier models its mean pass rate rises 106.5–181.9% over single-run baselines and 3.0–19.6% over native agent harnesses.

Recovering a partial differential equation from sampled solution data demands expert work in equation discovery, coefficient estimation and numerical solver configuration. RSI-PDEbench turns that workflow into a benchmark of 1,001 cases over 42 equation templates, split into noise-free, 50% Gaussian-noise and unseen-template-with-shifted-coefficient settings; the accompanying RSI-PDE agent shortlists candidate equations by residual, refines coefficients and solvers with relative feedback, and keeps improvements behind a validation gate. Across four frontier models its mean pass rate rises 106.5–181.9% over single-run baselines and 3.0–19.6% over native agent harnesses.

Physics-informed neural networks (PINNs) often lose accuracy on PDEs with sharp gradients, partly because the network struggles to represent high-frequency variation. CTPINN adds a learnable coordinate transformation alongside the solution network, mapping the physical domain to a computational domain where the solution is approximately affine and gradients are more evenly distributed. The explicit construction restores accuracy on boundary-layer problems, and the implicit construction, plugged into PirateNet, SPINN and QRPINN, improves accuracy across several benchmark equations.

Physics-informed neural networks (PINNs) often lose accuracy on PDEs with sharp gradients, partly because the network struggles to represent high-frequency variation. CTPINN adds a learnable coordinate transformation alongside the solution network, mapping the physical domain to a computational domain where the solution is approximately affine and gradients are more evenly distributed. The explicit construction restores accuracy on boundary-layer problems, and the implicit construction, plugged into PirateNet, SPINN and QRPINN, improves accuracy across several benchmark equations.

Automatic repair of PDE solvers now runs on a discrete diffusion language model: instead of rewriting the whole file left to right, it re-masks only the faulty code span and infills it with both left and right context visible. On five PDEBench families, with the same checkpoint, localized infilling raised average Pass@32 from 0.4938 to 0.5063 and cut per-attempt repair latency from 64.85 s to 0.748 s; from-scratch generation remains slower than same-scale autoregressive models, and worst-case end-to-end cost averages 55.5 s.

Automatic repair of PDE solvers now runs on a discrete diffusion language model: instead of rewriting the whole file left to right, it re-masks only the faulty code span and infills it with both left and right context visible. On five PDEBench families, with the same checkpoint, localized infilling raised average Pass@32 from 0.4938 to 0.5063 and cut per-attempt repair latency from 64.85 s to 0.748 s; from-scratch generation remains slower than same-scale autoregressive models, and worst-case end-to-end cost averages 55.5 s.

GLOBE compresses three-dimensional beam–field states into a compact latent space, learns the dominant evolution with a control-conditioned neural operator, and captures fine-scale residuals with a generative model, then applies parameter-free constraint operators after decoding. On two high-resolution 3D benchmarks, bunched-beam transport and plasma-wakefield acceleration, relative error is 9% and 13% lower than the strongest baseline DPOT, mid-frequency spectral error is 54% and 20% lower, and hours of particle simulation become about a minute per rollout.

GLOBE compresses three-dimensional beam–field states into a compact latent space, learns the dominant evolution with a control-conditioned neural operator, and captures fine-scale residuals with a generative model, then applies parameter-free constraint operators after decoding. On two high-resolution 3D benchmarks, bunched-beam transport and plasma-wakefield acceleration, relative error is 9% and 13% lower than the strongest baseline DPOT, mid-frequency spectral error is 54% and 20% lower, and hours of particle simulation become about a minute per rollout.