Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers

Aryan Chaudhary; Manikandan Padmanaban; Jagabondhu Hazra

cs.LGOct 3, 2026

Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers

Authors: Aryan Chaudhary, Manikandan Padmanaban, Jagabondhu Hazra

Organizations: BITS Pilani · IBM Research

Abstract

Operator eigenvalue problems appear throughout science. Classical methods usually discretize the operator into a matrix and then solve the resulting matrix eigenvalue problem. This works well in low dimensions, but fine grids quickly become expensive in both memory and computation as the dimension grows. Neural network based solvers avoid storing these large grids, but recent state of the art neural methods can require hundreds of thousands of training steps and may struggle to find the desired eigenvalues. We show that the two approaches can help each other. A coarse finite difference method (FDM) calculation acts as a cheap numerical model of the operator spectrum. We use the approximate eigenvalues as fixed shifts during the training of the neural solver, as they only need to locate the relevant part of the spectrum. We also introduce Stabilized Inverse Power Method Neural Network (SIPMNN), a more stable training procedure for higher-dimensional problems. Across five test problems at $d=10$ , the combined approach is more accurate overall than the tested fully neural alternatives while using eight to ten times fewer iterations.

Figures & tables

Panel of Figure 1 — Figure 1: SIPMNN first obtains rough eigenvalues from one coarse FDM solve. Their shifts stay fixed while the neural networks refine each mode. Learned deflation passes each lower-mode estimate to the higher-mode updates.

$d=5$
		IPMNN		SIPMNN
Benchmark	Reference $\lambda_{1}/\lambda_{2}$	Adaptive shift	Fixed FDM shift	Adaptive shift	Fixed FDM shift
Anharmonic	2.7957/4.0060	0.26 /1.17	0.43/4.02	2.20/12.21	0.78/ 0.77
Coupled oscillator	2.4597/3.3541	3.70/2.92	3.12 /238.40	137.61/60.42	8.00/ 1.10
Double well	4.3482/5.1403	0.10/ 0.11	0.11/ 0.11	0.05 /0.26	0.07/0.13
Quantum harmonic	2.5000/3.5000	3.14/5.63	2.45 /429.46	218.16/61.66	8.08/ 1.80

Table 1: Relative eigenvalue error (%) after 50,000 neural optimization steps, reported as

\lambda_{1}/\lambda_{2}

. The lowest error for each mode is bold. The four configurations compare two training procedures and two choices of spectral shift.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Benchmark	Domain	Potential $V(x)$
Anharmonic	$[-4,4]^{d}$	$\frac{1}{2}\sum_{j}x_{j}^{2}+0.1\sum_{j}x_{j}^{4}$
Coupled oscillator	$[-5,5]^{d}$	$\frac{1}{2}[0.8\\|x\\|^{2}+0.2(\mathbf{1}^{\top}x)^{2}]$
Double well	$[-2.5,2.5]^{d}$	$\sum_{j}(x_{j}^{2}-1)^{2}$
Quantum harmonic	$[-5,5]^{d}$	$\frac{1}{2}\\|x\\|^{2}$
Random Gaussian	$[0,1]^{d}$	Sum of eight Gaussian wells

Appendix

Table 2: Schrödinger benchmark operators and domains.

Benchmark	Coarse FDM	Fixed FDM SIPMNN
Anharmonic	20.90/15.65	1.38/1.81
Coupled oscillator	35.38/21.89	28.06/5.10
Double well	10.87/12.01	0.53/1.50
Quantum harmonic	37.07/23.79	26.23/12.34
Random Gaussian	2.26/4.28	0.00/0.01

Appendix

Table 3: Error of the coarse

n=5

FDM estimate and the final SIPMNN result at

d=10

, reported as

\lambda_{1}/\lambda_{2}

in percent.

Benchmark	Full	No smooth target	No delayed update	No sigma penalty
Anharmonic	1.38 / 1.81	5.99/1.82	8.09/5.02	161.21/24.88
Coupled oscillator	28.06/ 5.10	25.99/9.44	25.63 /10.61	124.92/5.14
Double well	0.53/1.50	1.21/ 0.46	0.10 /1.46	66.71/98.45
Quantum harmonic	26.23 /12.34	29.67/13.93	30.05/ 10.59	139.69/152.31
Random Gaussian	0.00 / 0.01	0.00 / 0.01	0.00 /23.07	0.01/0.02

Appendix

Table 4: Fixed FDM SIPMNN ablation at

d=10

. Entries are relative errors (%) for

\lambda_{1}/\lambda_{2}

. The lowest error for each mode is bold.

Benchmark	Residual $r_{1}/r_{2}$
Anharmonic	0.078/0.124
Coupled oscillator	0.046/0.318
Double well	0.057/0.209
Quantum harmonic	0.064/0.057
Random Gaussian	0.032/0.395

Appendix

Table 5: Normalized residuals

r_{1}/r_{2}

for Fixed FDM SIPMNN at

d=10

Component	Configuration	Purpose
Coarse FDM	Tensor grid with $n=5$ interior points per coordinate and $h=(b-a)/(n+1)$ on a nonperiodic interval $[a,b]$	Fixed shifts
Neural collocation	50,000 static Latin hypercube points sampled separately from the FDM grid	Training
Anharmonic and Double Well references	Independent one-dimensional FDM with 800 interior points and a second-order central difference scheme. The $d$ -dimensional values are sums of one-dimensional eigenvalues	Evaluation
Coupled Oscillator and Quantum Harmonic references	Analytic eigenvalues	Evaluation
Random Gaussian references	Independent Galerkin calculation in a sine basis	Evaluation

Appendix

Table 6: Numerical calculations used for shifts, training, and evaluation.

Setting	Value
Coarse operator	Second-order central-difference stencil
Coarse eigensolver	LOBPCG [ Knyazev, 2001 ] , tolerance $10^{-7}$ , at most 500 iterations
Random seed	0 for all random number generators
Random Gaussian potential	$V(x)=\sum_{\ell=1}^{8}a_{\ell}\exp(-\\|x-c_{\ell}\\|_{2}^{2}/0.15^{2})$ , with $c_{\ell}\sim\operatorname{Uniform}([0,1]^{d})$ and $a_{\ell}\sim\operatorname{Uniform}([-5,5])$
Boundary condition	Map coordinates to $\widetilde{x}\in[-1,1]^{d}$ and multiply the raw output by $\prod_{j}(1-\widetilde{x}_{j}^{2})$
Neural network architecture	$[d,40,40,40,40,1]$ at $d=5$ and

Appendix

Table 7: Implementation and training configuration.

Neural PDE Solvers

Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.

See all 272 papers

Neural networks for spectral optimization
Sep 28, 2026Alexis de Villeroché, Beniamin Bogosel, Stéphane Breuils +2Neural Surrogate Modeling Spectral Optimization
SFO: Learning PDE Operators via Spectral Filtering
Jan 23, 2026Noam Koren, Rafael Moschopoulos, Kira Radinsky +1PDE Surrogate Modeling PDE Operator Learning
Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University
Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation
May 21, 2026Xiaotian Liu, Shuyuan Shang, Xiaopeng Wang +2Neural PDE Solvers Spectral Bias
1Dartmouth College · 2CUHK Shenzhen · 3Lawrence Berkeley National Lab

Sep 28, 2026J/K move · Enter open · S save

Neural networks for spectral optimization

Alexis de Villeroché, Beniamin Bogosel, Stéphane Breuils, Dorin Bucur+1

Neural Surrogate Modeling Spectral Optimization

Open paper AskPDF arXiv

Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1% mean relative error on the first ten eigenvalues, compared with 4% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.

Abstract

Neural PDE Solvers

Explore similar work

Neural networks for spectral optimization

SFO: Learning PDE Operators via Spectral Filtering

Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation