Trainable Spline Representations for Physics-Informed Learning

Gianluigi Rozza; Giovanni Canali; Nicola Demo

REVIEW 3 major objections 3 minor 25 references

On four PDE benchmarks, a trainable tensor-product B-spline ansatz beats neural physics-informed networks with far fewer parameters.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 22:24 UTC pith:35BN7OEI

load-bearing objection Genuinely new and useful method; the comparison overclaims the architecture advantage because boundary treatment and hyperparameters are not matched. the 3 major comments →

arxiv 2607.15751 v1 pith:35BN7OEI submitted 2026-07-17 cs.LG cs.NAmath.NA

Trainable Spline Representations for Physics-Informed Learning

Giovanni Canali , Nicola Demo , Gianluigi Rozza This is my paper

classification cs.LG cs.NAmath.NA

keywords physics-informed learningB-splinestensor-product spline approximationcontrol coefficientsPINNPDE solvingboundary conditionsparameter efficiency

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Physics-Informed Splines (PI-Splines) replace the neural-network ansatz at the heart of PINNs with a tensor-product B-spline expansion whose control coefficients are the only trainable parameters. The paper argues that this compact, structured representation—trained with the same residual-based loss as PINNs—is a competitive and stable alternative to neural physics-informed architectures. Across four benchmark problems (Poisson, anisotropic Helmholtz, sharp-exponential, and acoustic wave), PI-Splines report the lowest mean absolute error among all tested models while using up to roughly two orders of magnitude fewer parameters. The method enforces Dirichlet boundary conditions strongly by fixing boundary control coefficients, evaluates derivatives analytically, and makes each parameter local and geometrically interpretable. The authors frame it not as a universal replacement, but as a complementary tool for moderate-dimensional problems with smooth or piecewise-smooth solutions.

Core claim

The paper's central discovery is that a tensor-product B-spline trial field with trainable control coefficients, optimized against the PDE residual, matches or outperforms neural PINN baselines on every benchmark tested. Clamped knot vectors let boundary control coefficients be fixed to satisfy homogeneous Dirichlet data exactly, removing the boundary penalty from the objective. Derivatives follow from B-spline basis formulas rather than automatic differentiation, and compact support gives each coefficient a local geometric role. The reported mean absolute error is lowest among all compared architectures on all four problems, with one to two orders of magnitude fewer parameters than the neur

What carries the argument

The central object is the tensor-product B-spline expansion u_C(x) = sum_i C_i B_i(x), where C is a tensor of control coefficients and each B_i is a univariate B-spline basis factor along one coordinate direction. The entries of C are the optimization variables of the physics-informed residual loss. Three structural properties do the work: clamped endpoint knots let boundary layers of C be frozen to satisfy Dirichlet conditions exactly; analytical B-spline derivative formulas express PDE residuals through scaled finite differences of C without automatic differentiation; and compact support ensures local geometric meaning and sparse evaluation. Smoothness is explicitly set by spline order and

Load-bearing premise

The load-bearing premise is that the evaluation protocol isolates the approximation architecture: PI-Spline orders and control-point counts were tuned per problem, boundary conditions were enforced strongly rather than weakly, and baselines used fixed architectures—if those differences, rather than the spline representation itself, drive the reported accuracy gap, the central claim weakens.

What would settle it

Repeat the four benchmarks giving every baseline the same per-problem grid search over architecture size and a hard boundary-constrained construction; if PI-Splines no longer achieve the lowest error on all tasks, the claim that the spline representation is responsible for the improvement is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If PI-Splines hold up beyond these benchmarks, structured spline trial spaces become a viable first choice for physics-informed forward problems with smooth or piecewise-smooth solutions.
Strong boundary enforcement removes boundary-loss weighting from the tuning burden, reducing one of PINNs' main training instabilities for compatible problems.
Analytical spline derivatives mean residual evaluation is not tied to the depth of a network, enabling simpler gradient computation and possible precomputation of basis matrices.
Parameter counts are transparently tied to knot resolution, so error is controlled by control-point spacing—the sensitivity analysis shows resolution, not order, dominates accuracy.
Savings in parameters do not automatically become savings in wall-clock time because basis evaluation carries its own cost; efficient evaluation matters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Because PI-Splines confine each coefficient to a local support region, they suggest a direct adaptive strategy the paper only sketches: refine control points where the local residual is large, yielding a self-guided mesh-refinement loop for physics-informed training.
The derivative-control-coefficient formula turns derivatives into scaled finite differences of C; for linear PDEs, minimizing the residual is a quadratic program in C, so a direct linear-algebraic solve or preconditioned iterative method could replace stochastic optimization entirely.
For non-rectangular domains, the same construction can be carried by NURBS control grids, combining exact geometry with physics-informed residual training; the paper names this direction but does not test it.
A clean architecture-isolating test would give every baseline the same per-problem hyperparameter search and hard boundary enforcement; without that, the reported accuracy gap may partly reflect tuning protocol rather than the spline space itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Genuinely new and useful method; the comparison overclaims the architecture advantage because boundary treatment and hyperparameters are not matched.

read the letter

Two things to know. The method is genuinely new: directly training tensor-product B-spline coefficients with the PINN residual loss, with strong Dirichlet BCs enforced by fixing boundary control coefficients. That is distinct from the spline-in-NN hybrids (PIKAN, Spline-PINN, Deep NURBS) cited in the paper. The writing is clear, and the authors are honest about the finite-dimensional approximation space and the resolution-limited nature of the method.

The paper does real experimental work: code is public, five independent runs per problem, analytic reference solutions, MAE on fixed grids, and a sensitivity analysis that identifies the number of control points as the dominant factor. The parameter efficiency is real—169 to 1520 trainable parameters versus tens of thousands for the neural baselines—and training times are competitive. For smooth, moderate-dimensional forward problems on simple domains, this is a plausible practical solver.

The soft spot is the comparison protocol. The abstract and Section 3.2 claim matched loss terms and a comparison that 'primarily reflects the choice of approximation architecture,' but that is not accurate. PI-Splines impose homogeneous Dirichlet BCs strongly by freezing boundary coefficients and dropping the boundary loss; the neural baselines all enforce BCs as weak penalties with tuned (or untuned) weights. That is a structural difference in the objective, and for PINNs the boundary penalty is often a dominant error source. A matched strong-BC neural baseline is needed before the accuracy gap can be attributed to the spline space. There is also hyperparameter asymmetry: PI-Spline order and control points were selected per problem from the grid search in Section 3.3, while the baselines used fixed, hand-picked architectures. This second confound is real but more minor; methods need tuning, and the baselines could have been tuned too.

The stress-test note is right on both counts. None of this sinks the paper. The existence claim—that a directly trained B-spline trial space can be competitive—is credible, and the boundary treatment is a legitimate feature of the method, but the magnitude of the advantage over neural baselines is not established by these experiments.

Who benefits: researchers in physics-informed learning who care about interpretable, parameter-efficient alternatives to PINNs. It deserves a serious referee; the comparison needs to be fixed but the contribution is solid enough to warrant that effort. I would engage with the paper and cite it as related work, and I would bring it to a reading group as a good case study in experiment design for method comparisons.

Referee Report

3 major / 3 minor

Summary. The paper proposes Physics-Informed Splines (PI-Splines), a method that replaces the neural-network ansatz of PINNs with a tensor-product B-spline expansion whose control coefficients are optimized via a residual-based loss. Boundary conditions that are compatible with the spline representation can be enforced strongly by fixing boundary control coefficients. The method is evaluated on four benchmark problems (Poisson, anisotropic Helmholtz, sharp-gradient exponential, and acoustic wave) against PINN, Fourier-feature PINN, and PIKAN baselines. The paper reports that PI-Splines achieve the lowest mean absolute error on all four benchmarks while using substantially fewer trainable parameters, and provides a sensitivity analysis over spline order and control-point count.

Significance. If the reported comparison is valid, the paper makes a useful contribution by demonstrating that a compact, structured spline trial space can outperform overparametrized neural architectures in physics-informed training, with lower parameter counts, analytical derivatives, and local geometric interpretability. The manuscript includes several strengths: five independent runs per problem, analytic reference solutions, publicly available code and reproduction scripts, and a sensitivity analysis of the proposed method's main discretization parameters. However, the central empirical claim—that the accuracy gap is due to the approximation architecture—is currently confounded by two protocol asymmetries that need to be resolved before the quantitative conclusions can be accepted.

major comments (3)

[Section 2.3, Section 3.1, Abstract] PI-Splines impose homogeneous Dirichlet conditions strongly by fixing boundary control coefficients and removing the boundary loss, while all neural baselines (Section 2.1) enforce them weakly as penalty terms. The loss objectives therefore differ structurally between methods. The Abstract's claim of 'matched ... loss terms' and Section 3.2's assertion that the comparison 'primarily reflects the choice of approximation architecture' are not supported. This is load-bearing because the reported MAE advantage could stem from exact boundary satisfaction rather than from the B-spline trial space. Please provide a matched comparison: add a weak boundary penalty to PI-Splines, enforce strong boundary conditions for the baselines where feasible, or report boundary-error metrics alongside interior MAE.
[Table 4, Appendix A, Section 3.3] The PI-Spline configurations (spline order, control points per dimension) were selected per problem from the Section 3.3 grid search, whereas the neural baselines used fixed, hand-picked architectures (two hidden layers of width 128; PIKAN width 32, with only the knot count varied). This asymmetry in hyperparameter optimization can inflate the reported gap: the PI-Spline entries in Table 1 correspond to tuned configurations, not to a default architecture. Please either perform a comparable hyperparameter search for the baselines, or fix PI-Spline hyperparameters a priori, and quantify how much of the observed advantage is due to per-problem tuning.
[Section 3.2, Conclusions] The statement that the comparison 'primarily reflects the choice of approximation architecture' is an overclaim given the two confounds above. All benchmarks are smooth manufactured solutions on rectangular domains with simple boundary geometry, so the experiments do not yet isolate the effect of compact support or locality. A direct ablation that holds boundary treatment and tuning protocol fixed across trial spaces is needed to support the paper's central claim that the structured spline representation itself drives the accuracy improvement.

minor comments (3)

[Abstract] The phrase 'matched governing equations, collocation sets, loss terms, and optimization procedures' is inaccurate as written, because the boundary loss term is removed for PI-Splines. Please qualify this claim after the major revisions.
[Section 2.3] The sentence 'in the time-dependent experiments considered in this work, this hard enforcement strategy led to less stable optimization' reports an observation without supporting data. Consider adding a small experiment or a reference.
[Table 1] The training-time comparison sometimes depends on implementation details (e.g., PIKAN grid updates). A note that the reported times are on a single GPU with the provided code would help reproducibility; the code link is given, which is good.

Circularity Check

0 steps flagged

No significant circularity: PI-Spline derivation is self-contained and validated against external analytic benchmarks.

full rationale

The paper's central claim is an empirical comparison, not a derivation that reduces to its inputs. PI-Splines parametrize the unknown field with a tensor-product B-spline expansion and train control coefficients by the standard PINN residual objective. The reported accuracy is measured against independent analytic solutions on fixed evaluation grids, so the results are externally falsifiable and not constructed from the fitted values. The only self-citations are to the PINA software framework [2] and to the authors' related NURBS work [17]; these are used for implementation and related-work context, not as load-bearing justification of the method's validity. The strong enforcement of homogeneous Dirichlet boundary conditions and the per-problem selection of spline order and control-point counts are experimental-design choices that could affect the fairness of the comparison with neural baselines, but they are hyperparameter and protocol issues, not circular reasoning: no fitted parameter is renamed as a prediction, and the analytic solutions used for evaluation do not depend on the spline configuration. The paper also states its main limitation explicitly: the approximation space is finite-dimensional for fixed knots and order, and accuracy is resolution-limited. This candid limitation statement further confirms that no hidden reduction to inputs is being presented. Therefore, no circular step meeting the evidentiary standard is present.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The method introduces no new physical entities. Its free parameters are the spline discretization choices (order, control-point count, knot distribution, and wave loss weights), all selected by the authors, plus the standard assumptions of residual-based PINN training on smooth benchmark PDEs.

free parameters (4)

spline order per benchmark = Poisson 6, Helmholtz 7, Exponential 7, Wave 6
Chosen per problem from the sensitivity grid search in Section 3.3; the central accuracy claims depend on these choices.
control points per dimension = Poisson 15, Helmholtz 25, Exponential 25, Wave 40
Chosen per problem from the grid search; this directly sets the size of the approximation space and the number of trainable parameters.
knot distribution = uniform
All experiments use uniform knot vectors; the paper does not test adaptive or problem-adapted knot distributions.
wave-problem loss weights = PDE residual 1, initial displacement 50, initial velocity 10, boundary 10
Hand-chosen weighting for the acoustic wave benchmark, applied to all methods; the comparison would likely shift under different weights.

axioms (5)

standard math Cox–de Boor recurrence and derivative formulas for B-splines, including partition of unity and endpoint interpolation for clamped knot vectors.
Used throughout Section 2.2 to define basis functions, derivatives, and strong boundary conditions.
domain assumption The four benchmark PDEs are well-posed and have smooth analytic solutions on rectangular domains.
The method's success depends on the solutions being representable by moderately small tensor-product spline spaces; the paper acknowledges this is not a universal approximator at fixed resolution.
domain assumption Minimizing the residual and boundary losses at sampled collocation points produces an accurate approximation to the PDE solution.
This is the standard PINN training assumption, carried over without new theoretical justification in Section 2.3.
domain assumption A rectangular tensor-product grid with uniform knots is an adequate approximation space for the selected benchmarks.
All experiments use box domains and uniform knots; the method's performance on non-rectangular domains or strongly anisotropic features is not tested.
ad hoc to paper Strong enforcement of homogeneous Dirichlet conditions by fixing boundary control coefficients is stable and does not distort the interior solution.
The paper reports that hard imposition of initial conditions for time-dependent problems was less stable (Section 2.3), so the same principle is not applied uniformly; the choice is empirical.

pith-pipeline@v1.3.0-alltime-deepseek · 12580 in / 10263 out tokens · 86080 ms · 2026-08-01T22:24:47.324525+00:00 · methodology

0 comments

read the original abstract

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.

Figures

Figures reproduced from arXiv: 2607.15751 by Gianluigi Rozza, Giovanni Canali, Nicola Demo.

**Figure 1.** Figure 1: Poisson problem. Left: PI-Spline prediction. Middle: reference solution. Right: pointwise absolute error. 3.1.2 Helmholtz problem The second benchmark is a Helmholtz problem with anisotropic frequency content, posed on Ω = [−1, 1] × [−1, 1], following [15]: ∆u(x, y) + k 2u(x, y) = f(x, y) ∀(x, y) ∈ Ω, u(x, y) = 0 ∀(x, y) ∈ ∂Ω, (3.1.2) where f(x, y) = k 2 − a 2 1π 2 − a 2 2π 2 sin(a1πx) sin(a2πy). We set … view at source ↗

**Figure 2.** Figure 2: Helmholtz problem. Left: PI-Spline prediction. Middle: reference solution. Right: pointwise absolute error. 3.1.3 Exponential problem We then turn to a manufactured elliptic problem on Ω = [0, 1] × [0, 1], constructed to produce a solution with rapidly varying exponential features: ∆u(x, y) + ∂u ∂x = f(x, y) ∀(x, y) ∈ Ω, u(x, y) = 0 ∀(x, y) ∈ ∂Ω, (3.1.3) where the forcing term is defined as f(x, y) = (y − … view at source ↗

**Figure 3.** Figure 3: Exponential problem. Left: PI-Spline prediction. Middle: reference solution. Right: pointwise absolute error. 3.1.4 Acoustic wave problem Finally, we consider a one-dimensional acoustic wave equation over the spatial domain Ω = [0, 1] and the time interval T = [0, 1]: ∂ 2u ∂t2 − 4 ∂ 2u ∂x2 = 0 ∀(x, t) ∈ Ω × T, u(x, 0) = sin(πx) + 1 2 sin(4πx) ∀x ∈ Ω, ∂u ∂t (x, 0) = 0 ∀x ∈ Ω, u(x, t) = 0 ∀(x, t) ∈ ∂Ω × T. (… view at source ↗

**Figure 4.** Figure 4: Acoustic wave problem. Left: PI-Spline prediction. Middle: reference solution. Right: pointwise absolute error. 3.2 Quantitative comparison A quantitative comparison across all benchmark problems is reported in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Accuracy–cost–complexity trade-off across benchmark problems. Each panel corresponds [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Mean absolute error (MAE) across spline configurations for all benchmark problems. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 4 linked inside Pith

[1]

S. J. Anagnostopoulos, J. D. Toscano, N. Stergiopulos, and G. E. Karniadakis. Residual-based attention in physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 421:116805, 2024

2024
[2]

Coscia, A

D. Coscia, A. Ivagnes, N. Demo, and G. Rozza. Physics-informed neural networks for advanced modeling.Journal of Open Source Software, 8(87):5352, 2023. 13

2023
[3]

A. Daw, J. Bu, S. Wang, P. Perdikaris, and A. Karpatne. Mitigating propagation failures in physics-informed neural networks using retain-resample-release (R3) sampling.ArXiv, abs/2207.02338, 2023

Pith/arXiv arXiv 2023
[4]

de Boor.A Practical Guide to Splines, volume Volume 27

C. de Boor.A Practical Guide to Splines, volume Volume 27. Springer, 01 1978

1978
[5]

Grossmann, U

T. Grossmann, U. Komorowska, J. Latz, and C.-B. Sch¨ onlieb. Can physics-informed neural networks beat the finite element method?IMA Journal of Applied Mathematics, 89, 05 2024

2024
[6]

A. D. Jagtap and G. E. Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.Communications in Computational Physics, 28(5):2002–2041, 2020

2002
[7]

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics- informed machine learning.Nature Reviews Physics, 3, 2021

2021
[8]

D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In3rd International Conference on Learning Representations, 2015

2015
[9]

Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljacic, T. Y. Hou, and M. Tegmark. KAN: Kolmogorov–arnold networks. In Y. Yue, A. Garg, N. Peng, F. Sha, and R. Yu, editors, International Conference on Learning Representations, volume 2025, pages 70367–70413, 2025

2025
[10]

L. D. McClenny and U. M. Braga-Neto. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474:111722, 2023

2023
[11]

Piegl and W

L. Piegl and W. Tiller.The NURBS Book. Springer, 11 1996

1996
[12]

Rahaman, A

N. Rahaman, A. Baratin, D. Arpit, F. Dr¨ axler, M. Lin, F. A. Hamprecht, Y. Bengio, and A. C. Courville. On the spectral bias of neural networks. InInternational Conference on Machine Learning, 2019

2019
[13]

Raissi, P

M. Raissi, P. Perdikaris, and G. Karniadakis. Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019

2019
[14]

Saidaoui, L

H. Saidaoui, L. Espath, and R. Tempone. Deep NURBS—admissible physics-informed neural networks.Engineering with Computers, 40:4007–4021, 2024

2024
[15]

C. Si, M. Yan, X. Li, and Z. Xia. Complex physics-informed neural network.Journal of Computational Physics, 553:114713, 2026

2026
[16]

F. Sun, Y. Liu, Q. Wang, and H. Sun. PiSL: Physics-informed spline learning for data-driven identification of nonlinear dynamical systems.Mechanical Systems and Signal Processing, 191:110165, 2023

2023
[17]

Tamburlin, G

F. Tamburlin, G. Canali, G. A. D’Inverno, N. Demo, A. Mola, and G. Rozza. Constraint-driven optimization and parametrization of industrial NURBS geometries via neural deformation field.ArXiv, abs/2606.07198, 2026

Pith/arXiv arXiv 2026
[18]

J. D. Toscano, V. Oommen, A. J. Varghese, Z. Zou, N. A. Daryakenari, C. Wu, and G. E. Karniadakis. From PINNs to PIKANs: recent advances in physics-informed machine learning. Machine Learning for Computational Science and Engineering, 1, 2024

2024
[19]

Wandel, M

N. Wandel, M. Weinmann, M. Neidlin, and R. Klein. Spline-PINN: Approaching pdes without data using fast, physics-informed hermite-spline CNNs.Proceedings of the AAAI Conference on Artificial Intelligence, 36:8529–8538, 06 2022

2022
[20]

S. Wang, B. Li, Y. Chen, and P. Perdikaris. PirateNets: Physics-informed deep learning with residual adaptive networks.ArXiv, abs/2402.00326, 2024

Pith/arXiv arXiv 2024
[21]

S. Wang, S. Sankaran, H. Wang, and P. Perdikaris. An expert’s guide to training physics- informed neural networks.ArXiv, abs/2308.08468, 2023. 14

Pith/arXiv arXiv 2023
[22]

S. Wang, Y. Teng, and P. Perdikaris. Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055– A3081, 2021

2021
[23]

S. Wang, H. Wang, and P. Perdikaris. On the eigenvector bias of fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 384:113938, 2021

2021
[24]

S. Wang, X. Yu, and P. Perdikaris. When and why PINNs fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

2022
[25]

Z. Wang, R. Romagnoli, S. Mowlavi, and Y. Nakahira. Physics-informed deep b-spline net- works.Transactions on Machine Learning Research, 2026. A Training details All models were trained using a common physics-informed pipeline, with the aim of isolating the effect of the approximation architecture. For each benchmark problem, the governing equation, collo...

2026

[1] [1]

S. J. Anagnostopoulos, J. D. Toscano, N. Stergiopulos, and G. E. Karniadakis. Residual-based attention in physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 421:116805, 2024

2024

[2] [2]

Coscia, A

D. Coscia, A. Ivagnes, N. Demo, and G. Rozza. Physics-informed neural networks for advanced modeling.Journal of Open Source Software, 8(87):5352, 2023. 13

2023

[3] [3]

A. Daw, J. Bu, S. Wang, P. Perdikaris, and A. Karpatne. Mitigating propagation failures in physics-informed neural networks using retain-resample-release (R3) sampling.ArXiv, abs/2207.02338, 2023

Pith/arXiv arXiv 2023

[4] [4]

de Boor.A Practical Guide to Splines, volume Volume 27

C. de Boor.A Practical Guide to Splines, volume Volume 27. Springer, 01 1978

1978

[5] [5]

Grossmann, U

T. Grossmann, U. Komorowska, J. Latz, and C.-B. Sch¨ onlieb. Can physics-informed neural networks beat the finite element method?IMA Journal of Applied Mathematics, 89, 05 2024

2024

[6] [6]

A. D. Jagtap and G. E. Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.Communications in Computational Physics, 28(5):2002–2041, 2020

2002

[7] [7]

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics- informed machine learning.Nature Reviews Physics, 3, 2021

2021

[8] [8]

D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In3rd International Conference on Learning Representations, 2015

2015

[9] [9]

Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljacic, T. Y. Hou, and M. Tegmark. KAN: Kolmogorov–arnold networks. In Y. Yue, A. Garg, N. Peng, F. Sha, and R. Yu, editors, International Conference on Learning Representations, volume 2025, pages 70367–70413, 2025

2025

[10] [10]

L. D. McClenny and U. M. Braga-Neto. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474:111722, 2023

2023

[11] [11]

Piegl and W

L. Piegl and W. Tiller.The NURBS Book. Springer, 11 1996

1996

[12] [12]

Rahaman, A

N. Rahaman, A. Baratin, D. Arpit, F. Dr¨ axler, M. Lin, F. A. Hamprecht, Y. Bengio, and A. C. Courville. On the spectral bias of neural networks. InInternational Conference on Machine Learning, 2019

2019

[13] [13]

Raissi, P

M. Raissi, P. Perdikaris, and G. Karniadakis. Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019

2019

[14] [14]

Saidaoui, L

H. Saidaoui, L. Espath, and R. Tempone. Deep NURBS—admissible physics-informed neural networks.Engineering with Computers, 40:4007–4021, 2024

2024

[15] [15]

C. Si, M. Yan, X. Li, and Z. Xia. Complex physics-informed neural network.Journal of Computational Physics, 553:114713, 2026

2026

[16] [16]

F. Sun, Y. Liu, Q. Wang, and H. Sun. PiSL: Physics-informed spline learning for data-driven identification of nonlinear dynamical systems.Mechanical Systems and Signal Processing, 191:110165, 2023

2023

[17] [17]

Tamburlin, G

F. Tamburlin, G. Canali, G. A. D’Inverno, N. Demo, A. Mola, and G. Rozza. Constraint-driven optimization and parametrization of industrial NURBS geometries via neural deformation field.ArXiv, abs/2606.07198, 2026

Pith/arXiv arXiv 2026

[18] [18]

J. D. Toscano, V. Oommen, A. J. Varghese, Z. Zou, N. A. Daryakenari, C. Wu, and G. E. Karniadakis. From PINNs to PIKANs: recent advances in physics-informed machine learning. Machine Learning for Computational Science and Engineering, 1, 2024

2024

[19] [19]

Wandel, M

N. Wandel, M. Weinmann, M. Neidlin, and R. Klein. Spline-PINN: Approaching pdes without data using fast, physics-informed hermite-spline CNNs.Proceedings of the AAAI Conference on Artificial Intelligence, 36:8529–8538, 06 2022

2022

[20] [20]

S. Wang, B. Li, Y. Chen, and P. Perdikaris. PirateNets: Physics-informed deep learning with residual adaptive networks.ArXiv, abs/2402.00326, 2024

Pith/arXiv arXiv 2024

[21] [21]

S. Wang, S. Sankaran, H. Wang, and P. Perdikaris. An expert’s guide to training physics- informed neural networks.ArXiv, abs/2308.08468, 2023. 14

Pith/arXiv arXiv 2023

[22] [22]

S. Wang, Y. Teng, and P. Perdikaris. Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055– A3081, 2021

2021

[23] [23]

S. Wang, H. Wang, and P. Perdikaris. On the eigenvector bias of fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 384:113938, 2021

2021

[24] [24]

S. Wang, X. Yu, and P. Perdikaris. When and why PINNs fail to train: A neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022

2022

[25] [25]

Z. Wang, R. Romagnoli, S. Mowlavi, and Y. Nakahira. Physics-informed deep b-spline net- works.Transactions on Machine Learning Research, 2026. A Training details All models were trained using a common physics-informed pipeline, with the aim of isolating the effect of the approximation architecture. For each benchmark problem, the governing equation, collo...

2026