arxiv: 2605.02280 · v1 · submitted 2026-05-04 · 💻 cs.LG · cs.NA· math.NA

Recognition: 3 theorem links

· Lean Theorem

Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates

Xinyu Li , Jianhua Zhang , Liang Chen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:51 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA

keywords variational methodsFourier networksparametric surrogate modelingmultiphysics simulationphysics-informed neural networksmatrix learningPDE solverssurrogate models

0 comments

The pith

A Fourier neural network turns multiphysics PDE training into one linear matrix solve for fast parametric surrogates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a variational matrix-learning Fourier network (VMLFN) for building surrogate models of parametric multiphysics problems governed by PDEs. These problems require many repeated solutions during design exploration, which is costly with standard finite-element methods. The VMLFN fixes the hidden-layer parameters using a log-space sine representation and randomly sampled frequencies, then solves directly for the output weights by enforcing the stationarity condition on a variational energy functional derived from the PDE weak form. This converts the usual iterative physics-informed training into a single linear matrix problem that uses only first-order derivatives. Tests on heat conduction, solid mechanics, and wave propagation show accurate full-field results with large speedups over both conventional neural networks and repeated simulations.

Core claim

VMLFN constructs a log-space sine neural representation with randomly sampled spectral frequencies, frequency-dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden-layer parameters, the output-layer weights are determined by reformulating the governing PDEs into variational weak forms and enforcing the stationarity condition of the resulting energy functional. This converts physics-informed training into a linear matrix-solving problem, requiring only first-order derivatives and avoiding both high-order automatic differentiation and penalty-coefficient tuning. A heuristic frequency-scanning algorithm selects a problem-adaptive maximum frequency.

What carries the argument

The variational matrix-learning Fourier network, which fixes random spectral frequencies in a log-space sine hidden layer and solves for output weights via the stationarity condition of the PDE variational energy functional, turning training into a linear solve.

Load-bearing premise

The heuristic frequency-scanning algorithm selects a maximum frequency that covers the dominant spectral content, and fixed hidden-layer parameters with the log-space sine representation suffice to represent solutions accurately across the full parametric design space.

What would settle it

If finite-element reference solutions for a new parameter value within the design space show large pointwise errors in the VMLFN predictions, especially when the scanned frequencies miss key modes or for problems with sharp local features, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.02280 by Jianhua Zhang, Liang Chen, Xinyu Li.

**Figure 1.** Figure 1: Overall framework of the proposed variational matrix-learning Fourier network for parametric multiphysics surrogate modeling. view at source ↗

**Figure 2.** Figure 2: Log-space sine neural network. To enhance the spectral representation capability of the proposed framework, we develop a new Fourier-type neural architecture, referred to as the log-space sine neural network, as shown in view at source ↗

**Figure 3.** Figure 3: Numerical approximation of a one-dimensional integral. (a) view at source ↗

**Figure 4.** Figure 4: Heuristic scanning algorithm for maximum frequency selec view at source ↗

**Figure 5.** Figure 5: Wave-field distributions for the original polynomial manufactured solution on the representative slice view at source ↗

**Figure 6.** Figure 6: Wave-field distributions for the high-frequency cosine–cosine–sine manufactured solution on the representative slice view at source ↗

**Figure 7.** Figure 7: Temperature-field distributions for the Gaussian heat-source problem on two representative cross-sectional slices. (a)–(e) Results view at source ↗

**Figure 8.** Figure 8: Temperature-field distributions for the actual chip heat-source problem on two representative cross-sectional slices. (a)–(e) Results view at source ↗

**Figure 9.** Figure 9: Temperature-field distributions for the heat-source problem on the representative slice view at source ↗

**Figure 10.** Figure 10: Temperature-field distributions for the heat-source problem on the representative slice view at source ↗

**Figure 11.** Figure 11: Temperature-field distributions for the heat-source problem with view at source ↗

**Figure 12.** Figure 12: Temperature-field distributions for the heat-source problem with view at source ↗

**Figure 13.** Figure 13: Temperature-field distributions for the heat-source problem with view at source ↗

**Figure 14.** Figure 14: Warpage distributions of the 3-D thermoelastic deformation field in the heterogeneous composite, represented by the out-of-plane view at source ↗

read the original abstract

Multiphysics simulation is critical for system-technology co-optimization (STCO) in chiplet-based design, but repeated finite-element solutions of PDE-governed problems are computationally expensive in parametric design exploration. This paper proposes a variational matrix-learning Fourier network (VMLFN) for efficient parametric multiphysics surrogate modeling. VMLFN constructs a log-space sine neural representation with randomly sampled spectral frequencies, frequency-dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden-layer parameters, the output-layer weights are determined by reformulating the governing PDEs into variational weak forms and enforcing the stationarity condition of the resulting energy functional. This converts physics-informed training into a linear matrix-solving problem, requiring only first-order derivatives and avoiding both high-order automatic differentiation and penalty-coefficient tuning. A heuristic frequency-scanning algorithm is further introduced to select a problem-adaptive maximum frequency that covers the dominant spectral range of the target problem. The proposed method is validated on heat conduction, solid mechanics, and Helmholtz wave propagation problems. Results from five benchmark cases demonstrate that VMLFN delivers accurate full-field predictions with substantial speedup over conventional physics-informed neural networks and repeated finite-element simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

VMLFN converts physics-informed training to a linear solve on a fixed log-sine basis chosen by heuristic scan, but the abstract gives no numbers or parametric spectral checks to show the basis holds up.

read the letter

The paper's main move is to fix the hidden-layer frequencies and decay rates in advance, then solve a linear system for the output weights by enforcing stationarity of a variational weak form. This replaces the usual iterative optimization in PINNs with a matrix solve that needs only first derivatives and no penalty tuning for boundaries. The log-space sine representation plus frequency-dependent decay is the specific choice they make for the basis functions, and a scanning procedure picks the maximum frequency per problem type. That combination is what they present as new for parametric multiphysics surrogates in heat conduction, elasticity, and wave problems. It does sidestep two common PINN pain points and could be faster when the basis is adequate. The abstract claims accurate full-field results and speedups on five benchmarks, but supplies none of the actual error values, timing tables, or baseline comparisons. The central risk is exactly the one the stress-test note flags: the basis is frozen while parameters vary, so any shift in the solution's frequency content outside the pre-scanned band leaves the linear solve projecting onto an incomplete space. Nothing in the provided description shows spectral plots, error curves versus parameter values, or tests that the heuristic scan remains robust across the design space. The frequency choice itself is problem-adaptive rather than parameter-free, which limits how general the surrogate can be without re-scanning. This work is aimed at engineers who need fast parametric sweeps for chiplet or multiphysics optimization and who already know variational methods or Fourier-feature networks. A reader looking for concrete alternatives to standard PINN training might try the linear-solve step on their own problems. The paper deserves a serious referee because the reformulation is testable and the speed claim is practically relevant, even though the current evidence is thin. I would send it out for review to get the quantitative results and any hidden implementation details examined.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the Variational Matrix-Learning Fourier Network (VMLFN) for parametric multiphysics surrogate modeling. It constructs a log-space sine neural representation with randomly sampled spectral frequencies, frequency-dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden-layer parameters, the output-layer weights are determined by reformulating the governing PDEs into variational weak forms and solving the resulting linear system from the stationarity condition of the energy functional. A heuristic frequency-scanning algorithm selects a problem-adaptive maximum frequency. The method is validated on five benchmarks involving heat conduction, solid mechanics, and Helmholtz wave propagation, with claims of accurate full-field predictions and substantial speedups over conventional physics-informed neural networks and repeated finite-element simulations.

Significance. If the central claims hold, VMLFN could provide an efficient alternative for parametric design exploration in multiphysics problems, reducing the cost of repeated simulations in applications such as chiplet-based STCO. The conversion of physics-informed training to a linear matrix solve is a notable technical strength, as it requires only first-order derivatives and eliminates penalty-coefficient tuning. The combination of random Fourier features with variational principles offers a potentially reproducible and stable framework, though its parametric robustness remains to be fully substantiated.

major comments (2)

The heuristic frequency-scanning algorithm (method section) selects a single fixed maximum frequency whose randomly sampled log-space sines, after decay regulation, are assumed to form an adequate basis for every point in the design space. Because hidden-layer parameters remain frozen while only output weights are solved per parameter instance, any parametric variation that shifts the dominant spectral content outside the pre-scanned band produces a Galerkin projection onto an incomplete subspace; the manuscript provides no spectral plots, error-vs-frequency curves, or cross-parameter validation to confirm robustness.
Results section (five benchmark cases): the claims of 'accurate full-field predictions' and 'substantial speedup' are central to the contribution, yet the abstract and summary provide no quantitative error metrics (e.g., relative L2 norms), baseline comparisons with specific numbers, or timing breakdowns; without these, the speedup and accuracy assertions cannot be evaluated against the stated alternatives.

minor comments (2)

Abstract: inclusion of at least one representative quantitative result (average error or speedup factor) would better support the performance claims for readers.
Notation: the precise mathematical form of the frequency-dependent decay regulation and its embedding into the sine basis should be stated explicitly with an equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments. We address each major comment point by point below and have revised the manuscript to incorporate additional evidence and quantitative details where the original presentation was insufficient.

read point-by-point responses

Referee: The heuristic frequency-scanning algorithm (method section) selects a single fixed maximum frequency whose randomly sampled log-space sines, after decay regulation, are assumed to form an adequate basis for every point in the design space. Because hidden-layer parameters remain frozen while only output weights are solved per parameter instance, any parametric variation that shifts the dominant spectral content outside the pre-scanned band produces a Galerkin projection onto an incomplete subspace; the manuscript provides no spectral plots, error-vs-frequency curves, or cross-parameter validation to confirm robustness.

Authors: We agree that explicit validation of basis adequacy across the parametric design space is necessary to support the fixed-hidden-layer approach. In the revised manuscript we have added spectral content plots for multiple parameter instances spanning the design space, error-versus-frequency curves demonstrating convergence within the scanned band, and cross-parameter validation results confirming that the heuristically selected maximum frequency remains sufficient for all tested instances. These additions substantiate that the Galerkin projection remains accurate without requiring per-instance frequency adaptation. revision: yes
Referee: Results section (five benchmark cases): the claims of 'accurate full-field predictions' and 'substantial speedup' are central to the contribution, yet the abstract and summary provide no quantitative error metrics (e.g., relative L2 norms), baseline comparisons with specific numbers, or timing breakdowns; without these, the speedup and accuracy assertions cannot be evaluated against the stated alternatives.

Authors: We acknowledge that the abstract and introductory summary lacked the specific quantitative metrics needed for immediate evaluation. The results section already contains relative L2 error norms, direct numerical comparisons against PINN and repeated FEM baselines, and wall-clock timing breakdowns for all five benchmarks. In the revision we have updated the abstract to include representative quantitative values (e.g., relative L2 errors and speedup factors) and added a consolidated summary table in the results section for clarity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in variational reformulation or linear solve

full rationale

The paper's derivation chain reformulates governing PDEs into variational weak forms whose stationarity yields a linear system for output-layer weights (with hidden-layer frequencies and decay rates held fixed). This is a direct application of standard Ritz-Galerkin methods and does not reduce the computed solution to the input data or parameters by construction. The heuristic frequency-scanning step selects a maximum frequency but remains an external preprocessing choice; the subsequent linear solve still enforces the weak-form residual and is not tautological. No self-citations, self-definitional loops, or fitted inputs relabeled as predictions appear in the abstract or described method. The overall procedure is an approximation whose accuracy is assessed empirically on benchmarks rather than guaranteed by the derivation itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The method rests on standard variational principles for PDEs but introduces a new network variant and a heuristic for frequency selection without external validation of the heuristic's robustness across problems.

free parameters (1)

maximum frequency
Selected by the heuristic frequency-scanning algorithm to cover the dominant spectral range of the target problem

axioms (1)

standard math Stationarity condition of the energy functional obtained from variational weak forms of the governing PDEs
Invoked to convert physics-informed training into a linear matrix-solving problem

invented entities (1)

Variational Matrix-Learning Fourier Network (VMLFN) no independent evidence
purpose: Efficient parametric multiphysics surrogate modeling
Newly proposed architecture combining log-space sine representation with variational matrix learning

pith-pipeline@v0.9.0 · 5511 in / 1354 out tokens · 73332 ms · 2026-05-08T18:51:03.092653+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost (Jcost, J(x) = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The basis function is defined as Ψ(x, θ) = D(x)η sin(2πω[x θ] + b) ... A frequency-dependent decay factor is introduced as η = 1/(1 + ‖2πω‖²)
IndisputableMonolith.Foundation.AlphaCoordinateFixation (log-coordinate calibration of cosh family) alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

log10 |ω| ∼ U(log10(ωmin), log10(ωmax)) ... allocates basis functions over a broad spectral range in a more balanced manner.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references

[1]

1.1 innovation for the next decade of compute eﬀiciency,

L. Su and S. Naffziger, “1.1 innovation for the next decade of compute eﬀiciency,” in Proc. IEEE Int. Solid-State Circuits Conf. (ISSCC), pp. 8–12, 2023

2023
[2]

Recent advances and trends in advanced packaging,

J. H. Lau, “Recent advances and trends in advanced packaging,” IEEE Transactions on Components, Packaging and Manufac- turing Technology, vol. 12, no. 2, pp. 228–252, 2022

2022
[3]

System-technology co- optimization for dense edge architectures using 3-d integration and nonvolatile memory,

L. M. Giacomini Rocha, M. Naeim, G. Paim, M. Brunion, P. Venugopal, D. Milojevic, J. Myers, M. Badaroglu, M. Ver- helst, J. Ryckaert, and D. Biswas, “System-technology co- optimization for dense edge architectures using 3-d integration and nonvolatile memory,” IEEE Journal on Exploratory Solid- State Computational Devices and Circuits, vol. 10, pp. 125–134, 2024

2024
[4]

Stco: Driving the more than moore era,

D. Biswas, J. Myers, S. B. Samavedam, and J. Ryckaert, “Stco: Driving the more than moore era,” in 2024 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), pp. 7–8, 2024

2024
[5]

Power and thermal integrity analysis of high performance and low power cpus at sub-2nm node designed with various advanced backside pdns,

L. Wang, F. Xie, J. Liu, T. Liu, L. Peng, Z. Zhang, T. Wei, and R. Chen, “Power and thermal integrity analysis of high performance and low power cpus at sub-2nm node designed with various advanced backside pdns,” in 2024 IEEE International Electron Devices Meeting (IEDM), pp. 1–4, 2024

2024
[6]

Stamp-2.5d: Structural and ther- mal aware methodology for placement in 2.5d integration,

V. D. Parekh, Z. W. Hazenstab, S. R. Srinivasa, K. Chakrabarty, K. Ni, and V. Narayanan, “Stamp-2.5d: Structural and ther- mal aware methodology for placement in 2.5d integration,” in 2025 IEEE 43rd International Conference on Computer Design (ICCD), pp. 159–166, 2025

2025
[7]

Accuracy- based hybrid parasitic capacitance extraction using rule-based, neural-networks, and field-solver methods,

M. S. Abouelyazid, S. Hammouda, and Y. Ismail, “Accuracy- based hybrid parasitic capacitance extraction using rule-based, neural-networks, and field-solver methods,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 41, no. 12, pp. 5681–5694, 2022

2022
[8]

More-stress: Model order reduction based eﬀicient numerical algorithm for thermal stress simulation of tsv arrays in 2.5d/3d ic,

T. Zhu, Q. Wang, Y. Lin, R. Wang, and R. Huang, “More-stress: Model order reduction based eﬀicient numerical algorithm for thermal stress simulation of tsv arrays in 2.5d/3d ic,” in 2025 Design, Automation & Test in Europe Conference (DATE), pp. 1–7, 2025

2025
[9]

Eﬀicient one-way wave-equation depth migration using fast fourier transform and complex padé approximation via helmholtz operator,

A. Li, “Eﬀicient one-way wave-equation depth migration using fast fourier transform and complex padé approximation via helmholtz operator,” IEEE Transactions on Geoscience and Remote Sensing, vol. 64, pp. 1–11, 2026

2026
[10]

Partial differential equations meet deep neural networks: A survey,

S. Huang, W. Feng, C. Tang, Z. He, C. Yu, and J. Lv, “Partial differential equations meet deep neural networks: A survey,” IEEE Trans. Neural Netw. Learn. Syst., vol. 36, no. 8, pp. 13649–13669, 2025

2025
[11]

PDE-net: Learning PDEs from data,

Z. Long, Y. Lu, X. Ma, and B. Dong, “PDE-net: Learning PDEs from data,” in Proceedings of the 35th International Conference on Machine Learning (J. Dy and A. Krause, eds.), vol. 80 of Proceedings of Machine Learning Research, pp. 3208–3216, PMLR, 10–15 Jul 2018

2018
[12]

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,

L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, “Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,” Nature Machine Intelligence, vol. 3, no. 3, pp. 218–229, 2021

2021
[13]

Fourier neural operator for parametric partial differential equations,

Z. Li, N. B. Kovachki, K. Azizzadenesheli, B. liu, K. Bhat- tacharya, A. Stuart, and A. Anandkumar, “Fourier neural operator for parametric partial differential equations,” in In- ternational Conference on Learning Representations, 2021

2021
[14]

Laplace neural operator for solving differential equations,

Q. Cao, S. Goswami, and G. E. Karniadakis, “Laplace neural operator for solving differential equations,” IEEE Trans. Neural Netw. Learn. Syst., vol. 6, no. 6, p. 631–640, 2024

2024
[15]

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, 2019

2019
[16]

Physics-informed machine learning,

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nat. Rev. Phys., vol. 3, no. 6, pp. 422–440, 2021

2021
[17]

An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, im- plementation and applications,

E. Samaniego, C. Anitescu, S. Goswami, V. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, and T. Rabczuk, “An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, im- plementation and applications,” Comput. Methods Appl. Mech. Engrg., vol. 362, p. 112790, 2020

2020
[18]

Physics-informed deep operator network for 3-d time-domain electromagnetic modeling,

S. Qi and C. D. Sarris, “Physics-informed deep operator network for 3-d time-domain electromagnetic modeling,” IEEE Trans- actions on Microwave Theory and Techniques, vol. 73, no. 7, pp. 3800–3812, 2025

2025
[19]

Seismic traveltime simulation for variable velocity models using physics-informed fourier neural operator,

C. Song, T. Zhao, U. Bin Waheed, C. Liu, and Y. Tian, “Seismic traveltime simulation for variable velocity models using physics-informed fourier neural operator,” IEEE Transactions on Geoscience and Remote Sensing, vol. 62, pp. 1–9, 2024

2024
[20]

Binn: A deep learning approach for computational mechanics problems based on boundary integral equations,

J. Sun, Y. Liu, Y. Wang, Z. Yao, and X. Zheng, “Binn: A deep learning approach for computational mechanics problems based on boundary integral equations,” Comput. Methods Appl. Mech. Engrg., vol. 410, p. 116012, 2023

2023
[21]

Advancing gener- alization in pinns through latent-space representations,

H. Wang, Y. Pu, S. Song, and G. Huang, “Advancing gener- alization in pinns through latent-space representations,” IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 12, pp. 20243–20257, 2025

2025
[22]

Bayesian neural networks with physics-informed priors with application to boundary layer velocity,

L. Menicali, D. H. Richter, and S. Castruccio, “Bayesian neural networks with physics-informed priors with application to boundary layer velocity,” IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 10, pp. 18048– 18061, 2025

2025
[23]

Monte carlo neural pde solver for learning pdes via probabilistic representation,

R. Zhang, Q. Meng, R. Zhu, Y. Wang, W. Shi, S. Zhang, Z.-M. Ma, and T.-Y. Liu, “Monte carlo neural pde solver for learning pdes via probabilistic representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 47, no. 6, 2025

2025
[24]

Extended physics- informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for non- linear partial differential equations,

A. D. Jagtap and G. E. Karniadakis, “Extended physics- informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for non- linear partial differential equations,” Commun. Comput. Phys., vol. 28(5), 2020

2020
[25]

Conser- vative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems,

A. D. Jagtap, E. Kharazmi, and G. E. Karniadakis, “Conser- vative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems,” Comput. Methods Appl. Mech. Engrg., vol. 365, p. 113028, 2020

2020
[26]

Physics informed extreme learn- ing machine (pielm)–a rapid method for the numerical solution of partial differential equations,

V. Dwivedi and B. Srinivasan, “Physics informed extreme learn- ing machine (pielm)–a rapid method for the numerical solution of partial differential equations,” Neurocomputing, vol. 391, pp. 96–118, 2020

2020
[27]

Optimization free neural network approach for solving ordinary and partial differential equa- tions,

S. Panghal and M. Kumar, “Optimization free neural network approach for solving ordinary and partial differential equa- tions,” Engineering with Computers, vol. 37, no. 4, pp. 2989– 3002, 2021

2021
[28]

Solving partial differential equation based on bernstein neural network and extreme learning machine algorithm,

H. Sun, M. Hou, Y. Yang, T. Zhang, F. Weng, and F. Han, “Solving partial differential equation based on bernstein neural network and extreme learning machine algorithm,” Neural Processing Letters, vol. 50, no. 2, pp. 1153–1172, 2019

2019
[29]

Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations,

S. Dong and Z. Li, “Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations,” Comput. Methods Appl. Mech. Engrg., vol. 387, p. 114129, 2021

2021
[30]

Numerical solution and bifurcation analysis of nonlinear partial differential 19 equations with extreme learning machines,

G. Fabiani, F. Calabrò, L. Russo, and C. Siettos, “Numerical solution and bifurcation analysis of nonlinear partial differential 19 equations with extreme learning machines,” Journal of Scientific Computing, vol. 89, no. 2, p. 44, 2021

2021
[31]

Bayesian physics- informed extreme learning machine for forward and inverse pde problems with noisy data,

X. Liu, W. Yao, W. Peng, and W. Zhou, “Bayesian physics- informed extreme learning machine for forward and inverse pde problems with noisy data,” Neurocomputing, vol. 549, p. 126425, 2023

2023
[32]

An extreme learning machine-based method for computational pdes in higher dimensions,

Y. Wang and S. Dong, “An extreme learning machine-based method for computational pdes in higher dimensions,” Comput. Methods Appl. Mech. Engrg., vol. 418, p. 116578, 2024

2024
[33]

Bridging traditional and machine learning-based algorithms for solving pdes: The random feature method,

J. Chen, X. Chi, W. E, and Z. Yang, “Bridging traditional and machine learning-based algorithms for solving pdes: The random feature method,” Journal of Machine Learning, vol. 1, p. 268–298, Sept. 2022

2022
[34]

Least squares with equality constraints extreme learning machines for the resolution of pdes,

D. Elia De Falco, E. Schiassi, and F. Calabrò, “Least squares with equality constraints extreme learning machines for the resolution of pdes,” Journal of Computational Physics, vol. 547, p. 114553, 2026

2026
[35]

Neural network algorithm based on legendre improved extreme learning machine for solving elliptic partial differential equa- tions,

Y. Yang, M. Hou, H. Sun, T. Zhang, F. Weng, and J. Luo, “Neural network algorithm based on legendre improved extreme learning machine for solving elliptic partial differential equa- tions,” Soft Comput., vol. 24, p. 1083–1096, Jan. 2020

2020
[36]

Laguerre neural network for solving neutral delay differential equations based on the extreme learning machine,

W. Wang, J. Pan, F. Gao, H. Liu, and L. Zhou, “Laguerre neural network for solving neutral delay differential equations based on the extreme learning machine,” International Journal of Computer Mathematics, vol. 103, no. 2, pp. 357–378, 2026

2026
[37]

Numerical solution of elliptic pdes via vieta–fibonacci wavelet based neural networks,

S. Aeri and R. Kumar, “Numerical solution of elliptic pdes via vieta–fibonacci wavelet based neural networks,” Nonlinear Science, vol. 7, p. 100137, 2026

2026
[38]

Linear time electromigration analysis based on physics- informed sparse regression,

L. Chen, W. Jin, M. Kavousi, S. Lamichhane, and S. X.-D. Tan, “Linear time electromigration analysis based on physics- informed sparse regression,” IEEE Transactions on Computer- Aided Design of Integrated Circuits and Systems, vol. 42, no. 11, pp. 4126–4138, 2023

2023
[39]

Learning in sinusoidal spaces with physics-informed neural networks,

J. C. Wong, C. C. Ooi, A. Gupta, and Y.-S. Ong, “Learning in sinusoidal spaces with physics-informed neural networks,” IEEE Transactions on Artificial Intelligence, vol. 5, no. 3, pp. 985– 1000, 2024

2024
[40]

General fourier feature physics-informed extreme learning machine (gff- pielm) for high-frequency pdes,

F. Ren, S. Wang, P.-Z. Zhuang, H.-S. Yu, and H. Yang, “General fourier feature physics-informed extreme learning machine (gff- pielm) for high-frequency pdes,” 2025

2025
[41]

Physics-guided spectral neural networks with self-discovered partial differential equa- tions for sparse field reconstruction,

G. Xiao, Q. Lang, W. Lu, and X. Liu, “Physics-guided spectral neural networks with self-discovered partial differential equa- tions for sparse field reconstruction,” IEEE Transactions on Industrial Informatics, vol. 22, no. 3, pp. 1771–1781, 2026

2026
[42]

Fourier neural networks as function approximators and differential equation solvers,

M. Ngom and O. Marin, “Fourier neural networks as function approximators and differential equation solvers,” Statistical Analysis and Data Mining: The ASA Data Science Journal, vol. 14, no. 6, pp. 647–661, 2021

2021
[43]

Fast full-chip parametric thermal analysis based on enhanced physics enforced neural networks,

L. Chen, J. Lu, W. Jin, and S. X.-D. Tan, “Fast full-chip parametric thermal analysis based on enhanced physics enforced neural networks,” in 2023 IEEE/ACM International Conference on Computer Aided Design (ICCAD), pp. 1–8, 2023

2023
[44]

Physics-informed neural networks with hard constraints for inverse design,

L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, and S. G. Johnson, “Physics-informed neural networks with hard constraints for inverse design,” SIAM Journal on Scientific Computing, vol. 43, no. 6, pp. B1105–B1132, 2021

2021
[45]

Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations,

E. Schiassi, R. Furfaro, C. Leake, M. De Florio, H. Johnston, and D. Mortari, “Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations,” Neurocomputing, vol. 457, pp. 334–356, 2021

2021
[46]

Pisov: Physics-informed separation of variables solvers for full-chip thermal analysis,

L. Chen, W. Zhu, M. Tang, S. X.-D. Tan, J.-F. Mao, and J. Zhang, “Pisov: Physics-informed separation of variables solvers for full-chip thermal analysis,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 44, no. 5, pp. 1874–1886, 2025

2025