FastLSQ 0.1.3

Solving PDEs in one shot via Fourier features with exact analytical derivatives

FastLSQ

Solving PDEs in one shot via Fourier features with exact analytical derivatives.

FastLSQ is a lightweight PDE solver built around SinusoidalBasis, an analytical derivative engine for random Fourier features. For sinusoidal features phi_j(x) = sin(W_j . x + b_j), every derivative of every order admits an exact closed-form expression -- no automatic differentiation needed.

Linear PDEs are solved in a single least-squares step; nonlinear PDEs are solved via Newton-Raphson iteration with Tikhonov regularisation, 1/sqrt(N) feature normalisation, and continuation/homotopy.

Installation

pip install fastlsq

For development (includes testing and build tools):

git clone https://github.com/asulc/FastLSQ.git
cd FastLSQ
pip install -e ".[dev]"

Quick start

Solve a linear PDE in one line

from fastlsq import solve_linear
from fastlsq.problems.linear import PoissonND

problem = PoissonND()
result = solve_linear(problem, scale=5.0)

u_fn = result["u_fn"]
print(f"Value error: {result['metrics']['val_err']:.2e}")

Solve a nonlinear PDE

from fastlsq import solve_nonlinear
from fastlsq.problems.nonlinear import NLPoisson2D

problem = NLPoisson2D()
result = solve_nonlinear(problem, max_iter=30)

print(f"Converged in {result['n_iters']} iterations")
print(f"Value error: {result['metrics']['val_err']:.2e}")

Use the basis directly

import torch
from fastlsq.basis import SinusoidalBasis

basis = SinusoidalBasis.random(input_dim=2, n_features=1500, sigma=5.0)
x = torch.rand(5000, 2)

# Arbitrary mixed partial via multi-index
d2_dxdy = basis.derivative(x, alpha=(1, 1))

# Or use fast-path methods
H     = basis.evaluate(x)            # (5000, 1500)
dH    = basis.gradient(x)            # (5000, 2, 1500)
lap_H = basis.laplacian(x)           # (5000, 1500)

Compose PDE operators symbolically

import torch
from fastlsq.basis import SinusoidalBasis, Op

basis = SinusoidalBasis.random(input_dim=2, n_features=1500, sigma=5.0)
x = torch.rand(5000, 2)

# Coefficients can be scalars or nn.Parameter (for AdamW optimisation)
k, c = 10.0, 2.0
helmholtz = Op.laplacian(d=2) + k**2 * Op.identity(d=2)
A_pde = helmholtz.apply(basis, x)    # (5000, 1500)

wave = Op.partial(dim=2, order=2, d=3) - c**2 * Op.laplacian(d=3, dims=[0, 1])

Plot solutions

from fastlsq.plotting import plot_solution_2d_contour, plot_convergence

plot_solution_2d_contour(result["solver"], problem, save_path="solution.png")
plot_convergence(result["history"], problem_name=problem.name, save_path="convergence.png")

Benchmarks

# Linear PDE benchmark (Fast-LSQ vs PIELM)
python examples/run_linear.py

# Nonlinear PDE benchmark (Newton-Raphson)
python examples/run_nonlinear.py

# Learnable Helmholtz wavenumber (nn.Parameter + AdamW)
python examples/learnable_helmholtz.py

Inverse problems

The analytical derivatives enable gradients through the pre-factored solve, making inverse problems tractable. Example: recovering 4 anisotropic Gaussian heat sources (24 parameters) from 4 sparse sensors. The heat equation is solved in space-time; L-BFGS-B optimises source positions and shapes to match sensor time-series. (Click image for animation.)

python examples/inverse_heat_source.py

Core architecture

The framework is built around SinusoidalBasis -- the analytical derivative engine:

Class	Purpose
SinusoidalBasis	Evaluates basis functions and arbitrary-order derivatives in O(1) via the cyclic identity
BasisCache	Pre-computes sin(Z)/cos(Z) once, reuses across multiple derivative evaluations
DiffOperator / Op	Symbolic linear differential operators that compose via +, -, scalar *; coefficients can be nn.Parameter for learnable PDEs
FeatureBasis	Adapter for non-sinusoidal solvers (e.g. PIELM with tanh)
FastLSQSolver	Manages feature blocks; exposes .basis for all derivative computations
LearnableFastLSQ	Differentiable solver with learnable bandwidth via reparameterisation trick

How it works

1. Basis construction. Given collocation points x, construct a SinusoidalBasis with random weights W and biases b.

2. Analytical derivatives. Exploit the cyclic derivative identity: the n-th derivative of sin(z) cycles through {sin, cos, -sin, -cos} with monomial weight prefactors. Any mixed partial D^alpha phi_j(x) is computed in O(1) -- no computational graph, no automatic differentiation.

3. PDE assembly. Define the differential operator symbolically with Op (e.g. Op.laplacian(d=2)) and apply it to the basis to get the system matrix A.

4. Linear solve. Solve A beta = b via least squares (optionally Tikhonov-regularised).

5. Newton iteration (nonlinear). Linearise the PDE residual, solve J delta_beta = -R with backtracking line search, and repeat.

Adding your own PDE

Define a problem class and use solver.basis to build the linear system:

import torch, numpy as np
from fastlsq import solve_linear, Op
from fastlsq.geometry import sample_box, sample_boundary_box

class MyPoisson2D:
    def __init__(self):
        self.name = "My Poisson"
        self.dim = 2
        self.pde_op = -Op.laplacian(d=2)

    def exact(self, x):
        return torch.sin(np.pi * x[:, 0:1]) * torch.sin(np.pi * x[:, 1:2])

    def exact_grad(self, x):
        sx, cx = torch.sin(np.pi * x[:, 0:1]), torch.cos(np.pi * x[:, 0:1])
        sy, cy = torch.sin(np.pi * x[:, 1:2]), torch.cos(np.pi * x[:, 1:2])
        return torch.cat([np.pi * cx * sy, np.pi * sx * cy], dim=1)

    def source(self, x):
        return 2 * np.pi**2 * self.exact(x)

    def get_train_data(self, n_pde=5000, n_bc=1000):
        x_pde = sample_box(n_pde, self.dim)
        f_pde = self.source(x_pde)
        x_bc = sample_boundary_box(n_bc, self.dim)
        u_bc = self.exact(x_bc)
        return x_pde, [(x_bc, u_bc)], f_pde

    def build(self, solver, x_pde, bcs, f_pde):
        basis = solver.basis
        cache = basis.cache(x_pde)
        A_pde = self.pde_op.apply(basis, x_pde, cache=cache)
        As, bs = [A_pde], [f_pde]
        for (x_bc, u_bc) in bcs:
            As.append(100.0 * basis.evaluate(x_bc))
            bs.append(100.0 * u_bc)
        return torch.cat(As), torch.cat(bs)

    def get_test_points(self, n=5000):
        return sample_box(n, self.dim)

result = solve_linear(MyPoisson2D(), scale=5.0)

See examples/add_your_own_pde.py for the complete tutorial.

Features

• Analytical derivative engine: SinusoidalBasis computes arbitrary-order derivatives exactly in O(1) -- the foundation of the entire framework
• Symbolic PDE operators: Compose differential operators with Op (Laplacian, wave, Helmholtz, biharmonic, custom) via intuitive arithmetic; coefficients can be nn.Parameter for AdamW optimisation
• High-level API: Solve PDEs in one line with solve_linear() and solve_nonlinear()
• Learnable bandwidth: LearnableFastLSQ optimises the bandwidth (scalar or anisotropic) via reparameterisation
• Learnable PDE coefficients: Plug nn.Parameter into Op (e.g. Helmholtz wavenumber k) and optimise via AdamW; gradients flow through the prebuilt linear solve
• Auto-tuning: Automatic scale selection via grid search
• Built-in plotting: Solution visualization, convergence plots, spectral sensitivity
• Geometry samplers: Box, ball, sphere, interval, custom samplers
• Diagnostics: Problem validation, conditioning checks, error detection
• Export utilities: NumPy conversion, checkpoint saving/loading
• PyTorch Lightning: Integration for training loops
• 20+ benchmark problems: Linear, nonlinear, and regression-mode PDEs

Paper

The full preprint is available on arXiv

Citing this work

If you use FastLSQ in your research, please cite:

@misc{sulc2026fastlsqframeworkoneshotpde,
      title={FastLSQ: A Framework for One-Shot PDE Solving}, 
      author={Antonin Sulc},
      year={2026},
      eprint={2602.10541},
      archivePrefix={arXiv},
      primaryClass={math.NA},
      url={https://arxiv.org/abs/2602.10541}, 
}

License

This project is licensed under the MIT License -- see LICENSE for details.