Physics-Informed Deep B-Spline Networks
Authors: Zhuoyuan Wang
Raffaele Romagnoli
Saviz Mowlavi
Yorie Nakahira
Venue: Accepted by TMLR ()

Abstract: Physics-informed machine learning offers a promising framework for solving complex partial differential equations (PDEs) by integrating observational data with governing physical laws. However, learning PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains an open challenge. In this paper, we propose physics-informed deep B-spline networks, a novel technique that approximates a family of PDEs with different parameters and ICBCs by learning B-spline control points through neural networks. The proposed B-spline representation reduces the learning task from predicting solution values over the entire domain to learning a compact set of control points, enforces strict compliance to initial and Dirichlet boundary conditions by construction, and enables analytical computation of derivatives for incorporating PDE residual losses. While existing approximation and generalization theories are not applicable in this setting—where solutions of parametrized PDE families are represented via B-spline bases—we fill this gap by showing that B-spline networks are universal approximators for such families under mild conditions. We also derive generalization error bounds for physics-informed learning in both elliptic and parabolic PDE settings, establishing new theoretical guarantees. Finally, we demonstrate in experiments that the proposed technique has improved efficiency-accuracy tradeoffs compared to existing techniques in a dynamical system problem with discontinuous ICBCs and can handle nonhomogeneous ICBCs and non-rectangular domains.

Published in Transactions on Machine Learning Research (03/2026)
Physics-Informed Deep B-Spline Networks
Zhuoyuan Wang
zhwoyyuawrandree..cmm www. eduu
Deportment of Electrical and Computer Engineering Carnegie Mellon University
Raffaele Romagnoli
vomagnolir@duug edu
Department of Mathematics and Computer Science
Duquesne University
Saviz Mowlavi
mowlavi@mmrrl.com
Mitsubishi Electric Research Laboratories
Yorie Nakahira
yorie@cmmu.cd edu
Department of Electrical and Computer Engincering Carnegie Mellon University
Reviewed on OpenReview: https: // openreview. net/ forum?? id= t#02=Eqmzm
Abstract
Physics-informed machine learning offers a promising framework for solving complex partial differential equations (PDEs) by integrating observational data with governing physical laws. However, learning PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains open challenge. In this paper, propose physics-informed deep B-spline networks, novel technique that approximates lamily of PDEs with different parameters and ICBCs by learning B-spline control points through neural networks. The proposed B-spline representation reduces the learning task from predicting solution values over the entire domain to learning a compact set of control points, enforces strict compliance to initial and Dirichlet boundary conditions by construction, and enables analytical computation of derivatives for incorporating PDE residual losses. While existing approximation and generalization theories are not applicable in this setting- where solutions of parametrized PDE families represented via B-spline bases bases-we we fill this gap by showing that B-spline networks are universal approximators for such families under mild conditions. We also derive generalization error bounds for physics-informed learning in both elliptic and parabolic PDE settings, establishing new theoretical guarantees. Finally, we demonstrate in experiments that the proposed technique has improved afficiency-accuracy tradcoffs compared to existing techniques in a dynamical system problem with discontinuous ICBCs and can handle nonhomogeneous ICBCs and non-rectangular domains. Code available at http:://ghhubbbccmm//oowwwwggggg
1 Introduction
Recent advances in scientific machine learning have significantly accelerated progress in solving complex partial differential equations (PDEs). Physics-informed neural networks (PINNs) are proposed to combine information of available data and the governing physics model to learn the solutions of PDEs (Raissi et al. 2019 Han el al.) 2018). However, in dynamics scenarios, both the parameters of the PDE and those defining initial and boundary conditions (ICBCs) can vary over time. Consequently, efficiently solving PDEs across a broad range of parameter values becomes essential. Learning solutions for familics of parametric PDEs cspecialyy when ICBCs are highly variable or discontinuous remains a challenging task, requiring methods that are accurate, efficient, and theoretically sound. For example, in safety-critical control applications, time-varying

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Direct ICBC assignment
Physics Model
F. u)
Coefficient Network a)(u,a)
Physics Model Loss
Control Points
Prediction
Lp
C= (e
(x;
Data Loss
La
5 s|P
Analytical derivatives B B-Spline Basis
Training Data
B,(x,) ... Ba (x,)
Figure 1: Diagram of PI-BSNet. The coefficient network takes system and ICBC parameters input and outputs the control points tensor, which is then multiplied by the B-spline basis to produce the final output. Physics and data losses are used to train the coefficient network using closed-form derivative formulas, while the compliance of ICBC conditions is strictly enforced through the B-spline basis.
system dynamics and shifting safe regions result in continuously changing PDEs that characterize safcty probabilities (Wang et al., (2025). However, solving these PDEs in real time with limited onboard resources is often prohibitively costly. This motivates the need for neural network trained efficiently to represent family of PDEs that enables rapid online inference of solutions under varying parameters and 1CBCs.
Previous literature has generalized PINNs for parametric PDEs (Cho et al. 2024 Boudee et al. 2024 Huang et al. 2022), and operator learning methods have been developed to map input functions to PDE solutions (Kovachki et al., 2023: Li et al., 2020 Lu et al. 2019 These approaches typically impose initial and boundary conditions (ICBCs) soft constraints through penalty terms in the loss function, which often require extensive tuning and do not guarantee strict compliance (Son al. 2023 Brecht al.) 2023). Recent
works have introduced PINNs with hard ICBC constraints by explicitly constructing solution ansatzes that
satisfy the constraints by design (Wang et al. 2023 Li et al. 2024a Chen et al. 2023 Liu et al. 2022 Sun
let al. 2024). Such ansatzes are typically tailored to specific ICBCs and may lead to reduced accuracy in the interior aim to directly learn PDE solutions on the entire domain (Lu et al. 2019 Cuomo et al., 2022), without exploiting more compact or structured representations. In such approaches, the PDE residuals the training loss are computed by applying automatic differentiation over the entire space time domain, which can be computationalll--eet
Liu et al. (2022) Sun et al. (2024), EPINN in Fig. 2b. Moreover, most existing methods
In addition, while many approximation and generalization bounds derived for
PINNs and neural operators (Kovachki et al. 2023 Lu et al. 2019: De Ryck & Mishra 2022a b Mishra
& Molinaro) 2023 2022). theoretical results remain limited for families of PDEs represented using neural
networks with structured bases.
Contribution. In this work, we propose a novel framework, termed physics-infore informed deep B-spline networks (PI-BSNet), that integrates neural B-splines and physics-informed learning to jointly solve families of parametric PDEs with different ICBCs (Fig. D. Specifically, PI-BSNet is composed of B-spline basis functions and parameterized coefficient network that learns the weights for the B-spline basis. The coefficient network takes inputs of the PDE and ICBC parameters, and outputs a tensor of control points (i.e., weights for the B-spline basis), which are then multiplicd by the B-spline basis to generate the PDE values. This structured representation enforces compliance of ICBCs by construction. Training is performed using both physics-based and data- based loss functions to ensure the network accurately approximates PDE solutions across the domain for any PDEs in the parametric family. Furthermore, we show that the deep B-spline network serves a universal approximator for families of parametric PDEs under mild assumptions, and establish generalization error bounds for PI-BSNet. Finally, present experimental results
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showing that PI-BSNet has improved computation vs. accuracy trade-offs and is capable of handling nonhomogeneous ICBCs and non-reclangular domains. To the best of our knowledge, this is the first work to integrate B-spline basis representations with physics-informed learning for solving families of parametric PDEs while also providing theoretical guarantecs on approximation accuracy and generalization error bounds.
2 Related Work
Physics-Informed Learning. Physics-informed neural networks (PINNs) are neural networks that are trained to solve supervised learning tasks while respecting physics laws described by partial differential equations (Raissi et al. 2019 Han et al.
2018
Cuomo et al. 2022). These approaches have been widely
applied across domains, such as power systems
MMsyrrs et al. 2020j, fluid mechanics (Cai et al. 2022) and
medical applications (Sahli Costabal et al.
among others. Many variants of PINN have been developed to meet diverse learning requirements. For example, parametrized PINNs are proposed to solve parametric
2020
PDEs (Cho al. 2024 Huang al. 2022 de Avila Belbute-Pores al. 2021 Qin et al. 2022 Lei et al.
2024h. PINNs with hard constraints are proposed meet specific ICBC requirements (Wang et al.) 2023
Li et al. 2024a Chen et al.) 2023) PINN with adaptive rescaling or coupled differentiation schemes are
proposed to reduce the computation burden in training (Ko & Park 2025 Chiu et al., 2002)). Under certain
assumptions, PINNs are shown to exhibit bounded generalization error (De Ryck & Mishra 2022a
Mishra
& Molinaro
2023
2022) and convergence to ground truth solutions (Pang 2021
Pang el al.
2019
Jiao el al.
2021
Physics-informed neural operators (PINOs) are neural operators (Kovachki et al.
2023
Li ct al.
2020 Lu
et al.
2019
2021
2022) that learn mappings between functional spaces while respecting underlying physics
laws.
PINOs
have been applied to both fixed and parametric PDEs (Wang et al. 2021b Gao et al.
2021
et al.
2024b
Goswami et al. 2023), and extended to handle specialized problems such pattern formation
ct al.
2023 and varying ICBCs [Kmmrr et al. 2024). These methods typically impose ICBCs through loss
terms, which may not guarantee strict compliance (Brecht et al. 2023
In comparison, our method learns B-spline control point representations rather than solution values across the entire domain as in PINN PINN/PINO approaches (e.g.. Raissi et al. (2019): Han et al. (2018); Kovachki et al. (2023) ). Moreover, our method directly enforces ICBCs through the B-spline structure itself, rather than through loss functions (c.g.. Cuomo et al.
(2024b: : Goswami et al. (023)) problem-specific (2024a): Chen el al. (2023)). These architectural differences yield filters (e.g., Wang el al. )0 Li et al.
) Li ct al.
better compliance with boundary conditions (Fig.
2 stable and fast training (Fig. 3 and improved tradeoff
between computational efficioncy and prediction accuracy (Fig.
B-splines and neural networks. B-splines are piece-wise polynomial functions derived from slight. adjustments of Bezier curves, aimed obtaining polynomial curves that tie together smoothly (Ahlberg) ct al. 2016). B-splines have been integrated with finite element methods (Jia et al. 2013 Shen et al.) 2023
employed in variational dual formulations for PDEs (Sukumar & Acharra, 2024), used to parameterize PDE
domains (Falini et al. 2023), and adapted into spline-inspired mesh movement networks for PDEs (Song
et al. 2022).
The combination of B-splines with neural networks has produced diverse applications, inchuding surface reconstruction [Iglesias et al.] 2004), nonlinear system modeling (Yiu et al.] 2001 Wang et al.] (022b). image segmentation (Cho et al.) (2021), and control system design (Chen et al.) 2004]] Deng et al.] 2008
Golmogorov-Arnole Networks (KANs) [Luu et al.] 2024) employs spline functions to generate learnable weights alternative to traditional multilayer perceptrons. The neural network in our proposed P1-BSNet can take arbitrary MLP/ LaP//on--L--aaasee architectures, including KANs. In the regime of plysics-informed learning, convolutional neural networks (CNNs) with Hermite spline kernels are trained with PDE and boundary condition loss functions to provide forward-time prediction in 2022). Such method requires retraining when PDE parameters or boundary conditions change. In comparison, our work learns solutions on the entire state-time domain for parametric PDE families, and imposes hard compliance for varying TCBCs.
PDEs (Wandel et al,,
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Closest to work are methods that learn B-spline weights for paredpparaeter ODEs Fakhoory et al 2022
Romagnoli et al. 2024) and PDEs (Doleglo et al. 2022 Zhu et al. 2024). Our work extends these approaches in several key directions. First, while these works focus on learning a single PDE with fixed ICBCs, we focus on joint learning of a family of paramotric PDEs with different ICBCs with new thcoretical guarantees (Section 4.. Second, we leverage analytical derivatives of B-splines enable efficient physics-informed learning. The use of physics-informed learning with hard ICBCs compliance enables accurate generalization beyond regions with available training data (Theorem 4.6 and prediction on unseen parameters in Section
3 Proposed Method
3.1 Problem Formulation
The goal of this paper is to efficiently estimate high-dimensional surfaces governed by physics laws of wide range variety of parameters (e.g., the solution of a family of PDFs). We denote 8 R - R as the ground truth, ie., s(x) is the value of the surface at point I, where x E RR.. We assume the physics laws can be wrillen as
F(s, = 0, a & M(a),
B(s u) = O, a E 0l(().
where F is the physics law and B is the initial and boundary conditions (ICBCs), u C Rm is the parameters of the systems, M2(a)= E R" parameterized by A C Ø is the domain of interest. Here, U and A ranges for the system and domain paramcters and are bounded. In this paper, we consider n-dimensional bounded domain Ω = [a1. b1] x [a2, by] x ... x [an, I Our goal is to generate å with neural networks to estimate 8 on the entire domain of Ω with all possible parameters u and a. For example, in the case of solving 2D heat equations on (2) E [0, a]2 at time E [0, 10] with varying coefficient [0,2] 2] and a E [3, 4), we have
F(s,2,u) 0s/01- s00//00 8 8s/0z²) 0, B(8,I,u)
(2,, E fz S4 (21, T2, E 022 x S4
(2)
0,
(3)
where Ω2 = =00. al2 and Ω2 = [0, 10). and 02, is the boundary of Ωz Here, equation 2 is the heat equation
and iquation 3 is the boundary condition. In this case, we want to solve for 8 on = 22 x Q for all u E [0, 2 and a E [3, 4J. Similar problems have been studied in Li et al. (0024b) Gao et al. 2021); Cho et al. (2024 while the majority of the literature considers solving parameterized PDEs but with eit hir fixed coefficients or fixed domain and initial/ /oundary conditions. We slightly generalize the problem to consider systems with varying parameters, and with potentially varying domains and initial/boundie conditions.
3.2 B-Splines with Basis Functions
In this section, we introduce B-spline basis functions. We begin with the one-dimensional variable R. The B-spline basis functions are defined recursively by the Cox-di de Boor formula (Piegl & Tiller 2012):
8, Bz,d-1(z)+ + B144+1 F
B2,((1) = Eitd 2,
B141d-1(1),
(4)
214041 - 8:41
with the base case
11, 2, $
Br,(x)) - 0, otherwise.
(5)
Here, (x) denotes the value of the i-th B-spline basis function of order evaluated at r. The sequence (b.)(td+1 is a non-decreasing vector of knol poinls, where L is the number of B-spline basis functions. Since a B-spline is a piecewise polynomial function, the knot points determine the interval in which the polynomial is active.
Such domain configuration is widcly considered in the literature Takamoto et
2022
Gupta Brandstetter 2022
Li al. 2020 Raissi al. 2019 Wang al. 2021b Zhu al. 2024 Genera Generalizations lizations
considered in Section 3.3
Section
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There are multiple ways to choose knot points. In this work, we adopt clamped knot vectors, where the first and last knots are repeated d + times, i.e., d1 = .... ldt1 and 5z+1 knots equally spaced. For example, on the interval [0,3] with f= = 6 control points and order d = 3, the knot vector
Le+d+1, with the interior
[0,000000000012,,33333888
giving a total of + ++ 1== 10 knots.
The B-spline basis functions vector is defined as
Ba(x) = [B), (.. Br,)),,,,,,,,(..)) (z,((x))"
(6)
and the coefficients for these basis functions, namely control points, are defined as c:= [e1, C2,.... approximate solution s(x) with ((x) cBa(z). Note that with our choice of knot points, ensure the initial and final values of 8(x) coincide with the initial and final control points Ci and will be used later to directly impose initial conditions and Dirichlet boundary conditions with PI-BSNet. A visualization of B-spline basis functions and reconstructions can be found in Fig. in the Appendix.
ce]. , Then,
This property
More generally, for a n-dimensional space x = [=1.-. ... , 2n] E R", we can generate B-spline basis functions based on the Cox-de Boor recursion formula along each dimension 2; with order d, for 1,2, ... n, and the n-dimensional control point tensor will be given by C' [Ci,, J, xxxx where i and la are the index and the number of control points along the k-th dimension. We can then approximate the n-dimensional surface with B-splines and control points via
8 T2, Fnn
Gi in Bi (..)... Bin (In)-
3.3 Physics- Infermee B-Spline Networks
In this section, we introduce our proposed physics-informed deep B-spline networks (PI-BSNet). The overall diagram of the network is shown in Fig. [ The network composites a coefficient network that learns the control point tensor C' with system parameters u and ICBC parameters a, and the B-spline basis functions Bd; of order de for i = 1, .n. n. We use Gø(u, (x) to denote the PI-BSNet parameterized by 0, where (u,a) a:) is the input to the coefficient net, and is the input to the B-spline basis. We use C Go a) to denote the control points output by the coefficient network and 8(x) == Gø(u, a)()) to denote the PI-BSNet prediction. During the forward pass, the control point tensor C output from the coefficient net is multiplied with the B-spline basis functions Ba, via equation 7 to get the approximation 8. For the backward pass, two losses are imposed to efficiently and effectively train PI-BSNet. We first impose a physics model loss
Cp c==0 ZEP |
F(5,
where is the governing physics model of the system as defined in equation D and P the set of points sampled evaluate the governing physics model. When data available, we can additionally impose a data loss
C
Σ [s(z) - 3(x)) RED
(9)
D
to capture the mean square error of the approximation, where is the data point for the high dimensional surface, D is the set of points where data are available, and ă is the prediction from the P1-BSNet. Only for special types of boundary conditions that involve derivatives of the solution (e.g., Neumann and Robin types), we impose the following ICBC loss
C, = Σ 1 B(E,
(10)
IM
ZEM
where M is the set of points sampled to evaluate the 1CBC residual. The total loss is given by
L. Wyp wafad + wrLs,
(11)
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where Up. Wd and Wy the weights for physics, data and ICBC losses, and are usually set to values close to Detailed procedures for training PI-BSNet is shown in Alg. | in the Appendix. The choice of the loss function in equation [1 follows the standard formulation used in plysics-informed neural networks (Raissi ct al. 2019 Karniadakis et al., 2021). Detailed discussions on the effects, failure modes, and alternative
designs of such loss functions can be found (Wang al. 2021a Krishnapriyan et al. 2021
Note that several good properties of B-splines are leveraged in PI-BSNet.
First, advantageous training efficiency can be obtained with the B-spline representation. Specifically, the B-spline basis functions are fixed and can be calculated in advance, and the coefficient network is trained to learn only the fixed number of B-spline weights instead of the solution values on the entire space. This speeds up training over existing methods for certain problems.
Besides, any Dirichlet boundary conditions and initial conditions can be directly assigned via the control points tensor without any learning involved. This is a natural property of the B-spline representation ion with clamped knot. points (Ahlberg et al., 2016). This feature greatly enhances the accuracy of the learned solution near the ICBC, and improves the ease of design for the loss function weight factors are often used to impose stronger ICBC constraints in previous literature (Wang et al. 2022a).
Lastly, the derivatives of the B-spline functions can be analytically calculated. Specifically, the p-th derivative of the d-th ordered B-spline at arbitrary point is given by (Butterfield] 1976
d' Bia(#) drp
(d (d-1)) [d [d-
Beckdd-(a))
1)6
(12)
=== (#+td-s-1 Ritk)
Given this, we can directly calculate derivatives for the back-propagation of physies model loss Lp, which improves both computation efficiency and accuracy over numerical methods.
4 Theoretical Analysis
In this section, we provide theoretical guarantees for PI-BSNet. Despite integrating structured B-spline representation with physics-informed learning, we show that the proposed architecture remains a universal approximator for families parametric PDEs under mild regularity assumptions. Furthermore, we establish generalization error bounds for learning solution operators associated with clliptic and parabolic PDEs. To the best of our knowledge, such generalization guarantees for parametric PDE families have not been previously studied. All theorem proofs provided in Appendix A
4.1 Universal Approximation
In this section, we show that the proposed P1-BSNet is a universal approximator for solutions families of PDEs at arbitrary dimension. We consider n Hilbert spaces L2([a,, 6,]) for 1,2, ,.
Assumption 4.1. The solution of the physics problem defined in equation [ is continuous in a and u. Specifically, let 81 and 82 be the solutions of the physics problem with parameters an, un and a2, U2. For any e 0, there exist 81 and 82 > such that given ||o - a2/12 < 61, and ||u u2lla 82, we have ||s| -2122 Assumption 4.2. The solution of the physics problem defined in equation I is differentiable in
Assumption 4.1 is a basic assumption for a neural network to approximate solutions of families of parameterized PDEs, and not strict as it holds for many PDE problems 4 Assumption 4.2 holds for many PDE problems (Chen et. al.) 2018 De Angelis 2015 Barles et al. 2010), and our theoretical results can be generalized to cases where the solution is not differentiable at finite number of points.
2 AAblation experiments the effects of weights for physics and data losses be found in Appendix E.5 Uuder necessary domain mapping when 0n z ag. *For well-posed and stable PDE system with unique solution (e.g., linear Poisson, convection-diffusionn and heat equations with appropriate ICBCs), change of the system parameter or the ICBC parameter a usually results slight change of the value of the solution
Treves 1962).

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Theorem 4.3. Assume Assumption 4.1 and 4.2 hold. For any E N+ dimension, any and in a finite Then for any d-time.
parometer sel, let. de be the order of B-spline basis for dimension = 1,2,... differentiable function 22,1,22.-.. ,In) E [a(aa,bb] x (az, ba] x .... x [an. bnl) with d 2 max[d], dn where the domain depends on a and the function depends on u, and any e > 0, there exist a PI-BSNet configuraiinn Gg with enough width and depth, and corresponding poramelers 0* independent of u and a such that
1 c.
(13)
where 3 (u, the B-spline approrimation defined in equation 7 with the control points tensor Gø (u.
Theorem 4.3 tells us that the proposed PI-BSNet is a universal approximator of rrbitrary-dimensional surfaces with varying parameters and domains. Thus we know that when the solution of the problem defined in equation is unique, and the physics-informed loss functions Lp is densely imposed and altains zero (De Ryck & Mishra 2022a Mishra & Molinaro 2023),, we learn the solution of the PDE problem of arbitrary dimensions. The proof integrates individual universal approximation properties of neural networks (Hornik et al. 1989 Leshno et al. 1993) and B-splines (Blu & Unser 1999) for the proposed framework.
4.2 Generalization Error Bounds
In this section, we provide generalization error bounds of the proposed PI-BSNet for families of elliptic or parabolic PDEs. We make the following assumption about the Lipschitzness of the coefficient network and the training scheme of PI-BSNet.
Assumption 4.4. In the PI-BSNet framework, the output of the coefficient network is Lipschitz with respect to its inputs. Specifically, given the coefficient network G9 a), Vun, U2 such that ||u wall2 I 6us and Ya,, 02 such that ||a a2l12 & da, we have [[G((1,,a) - Ge (2))22 I L(6, + 6a), for some constant L. The training of PI-BSNet is finite subset of Utrain E and Auain E A for and a, respectively. The maximum interval between the samples in Utrain and Atrain is Au and Aa, and Urrain Acrain each fully covers U and A, i.e., Yuy e U and E A. there exists U2 E Uerain and ag E Arrain such that ||u1 - w2ll2 v Au, ||n - 022 2 S Aa.
Assumption 4.5.
Assumption 4.4 holds in practice neural networks are usually finite compositions of Lipschitz functions, and its Lipschitz constant can be estimated efficiently (Fazlyab et al. 2019). Assumption 4.5 can be easily achieved since one can sample PDE parameters and with equispaced intervals less than Au and Aa in U and A for training. We then have the following theorem to bound the generalization error for PI-BSNet on the family of elliptic or parabolic PDEs.
Thcorem 4.6. Assume Assumption
Assumption 4.4 and Assumption 4.5 hold. For any elliptic parabolic PDE with varying parumelers u E U and E A with U and A bounded, suppose that the domain of the PDE ((a) E R" is bounded, Su,a E (f((a n (?)()) is the solution where Ω is the closure of Ω F(su.a, 2) = 0, x. E I(a)) defines the PDE, and B( x) = 0, E (4) is the boundary condition. Let Gg denote a PI-BSNet parameterized by 0 and šu,a = Go (u, a) the solution predicted by PI-BSNet. Let Unif(.) denole the uniform distribution. If the following conditions holds:
1. (z) [IB(Go(u, 81, for all E Utrain and E Atruin- 2. (Z)(ni((())) [IF(G((u, x)]] 82, for all E Usran and Atrain
Ga(u,a), F(Ggu,,)) s(u, ai) are Lipschite continuous on M(a), for all and a E A.
Then for any u E U and A, the prediction error of Bu,a over ②(a) is bounded by
sup r(((a)
(z) sx.a(x)) < 81 + Md₂ + İ(Au + Aa),
(14)
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PI-BSNet (Proposed)
EPINN
VS-PINN
Pl-DeepONet
PI-ENO
1.0
1.0
10
Ground Truth
0.5
0.5
0.5
0.5
05
0
0.0
0.0
0.0
0
10
10
10
10
10
10
10
110
5
10
5
-10
0
t
0.0
10
1,0
1.0
1,0
1.0
0.5
0.5
0.5
05
0.5
0.0
0.0
0.0
0.0
00
10
10
10
-10
10
10
5
10
10
5
2
0
0
0
t
Figure 2: Recovery probability visualizations. Predictions in first and errors in second row.
where M is a constant depending on porameter sets A, u, domain functions Q2 lo. and the PDE F, L is some Lipschitz constant, and
28 (26(aa))
66 (o](ap(a)] T( ((tt))
81 max
21
R (o))))0aaa)))
(Ra,(a) (1122)/2 11/2
(15)
252 Ra(a)) Ra(a)19((a)
21 (8 [2][((a)): T(n/2 1) mtI
82 = max
(R xn/2
where Rey is the regularity of (.), 10)1 denotes the Lebesgue measure and T is the Gamma function.
Theorem 4.6 shows that PI-BSNet can generalize reliably across famiiles of elliptic and parabolic PDEs, rather than only matching the training instances. It establishes that small physics and boundary residuals well-covered set of training parameters unseen parameler values. The result makes explicit how generalization depends on the training coverage of the parameter space and on the smoothness of the learned mapping, thereby offering principled explanation for generalization across varying PDE parameters and boundary conditions without retraining, Moreover, the proof technique is not specific to the PI-BSNet architecture and can potentially be extended to other network architectures, for which comparable generalization error bounds have not yet. been systematically studied.
sufficient to guarantee uniformly small prediction errors for
5 Experiments
In this section, we present simulation results on estimating the recovery probability of a dynamical system which gives irregular ICBCs, and compare the proposed PI-BSNet with several baseline methods show advantages. We then present results show that PI-BSNet can handle nonhomogeneous 1CBCs and learn PDEs on non-roctangular trapezoid domains. All experiment details, ablation experiments and additional experiments can be lound in Appendix D and respectively.
5.1 Recovery Probabilities
We consider autonomous system with dynamics dz, dt + dwr, where x E R is the state, w, E IR is the standard Wiener process with W/ = 0, and R is the system parameter. Given a set C, = {I E R r 2 a), we want to estimate the probability of reaching C, at least once within time horizon starting at some To- Here, a the varying parameter the set C.. Mathematically this can be written as
sxxo = (Er E (), s.t. Ir C., mn).
(16)
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Data Loss
PDE Loss
101
10a
100
10.
10
10"
10'
10-1
CRAM
10-
SMACIIM
105
10
V5-995
CGN Fao
10
10"0-
10
10-4
10a4
1000
2000
4000
5000
1000
2000
5000
1000
2000
4000
5000
Figure 3: Losses vs. epochs with mean and standard deviation over 10 independent runs.
From Chern et al. cerlain initial and boundary conditions
2021 we know that such probability the solution of convection-diffusion equations with
PDE: ds
Ôs
pg 2 d (z, (z,1) -0, V(r C. x T
01 (x, 1) - dz
(x, 1)
(17)
ICBC:
s(a,t) = 1, 4t E ET. s(x, 0) = 0, Vx E C.
(18)
where CS the complement of Ca, and T |0, T] with T 10 be the time horizon of interest. Note that the initial condition and boundary condition at (a, (a, 0) is not continuous 5 which imposes difficulty for learning the solutions.
We train PI-BSNet with 3-layer fully connected neural networks with ReLU activation on varying parameters u E [0, 2] and a E [0,4] with both data and physics losses [7] and lest on randomly selected parameters in the same domain. We compare PI-BSNet with the standard physics-informed informed neural network (PINN) (Raissi et al, 2019),
PINNs that enforces hard constraints for 1CBCs including EPINN (Wang et al. 2023), SFHCPINN (Li 2024a). HWPINN (Chen al.) 2023), efficient PINN methods including VS-PPNNN (Ko & Park 2025)
ct al.
CAN- PINN
(Chiu et al. 2022), physics-iffrrmed neural operator methods including PI-DeepONet (Goswami
et al.
2024b) with similar comparable NN configurations. All methods are implemented to take PDE and ICBC parameters as additional inputs for parametric PDEs. All comparison
2023 PI-FNO (Li et al.
experiments are run a Linux machine with Intel i7 CPU and Nvidia GeForce RTX 4090 GPU. Details of
the experiment configuration can be found in Appendix D
Time Error Tradeoff
Fig. 2 visualizes the prediction results on a parameter set (u, ox)
from the same distribution but not used for training (see Fig. 12
Avg. Relative L2 Error 10-2
PI-NOO
in Appendix for full visualization results). We can see that PI-BSNet predicts the ground truth solution accurately during
SFHOPINN
testing while the other methods fails to do so. Fig. 3 visualizes the losses vs. epochs, and we can see that the loss for PI-BSNet drops the fastest and reaches convergence in the shortest amount of time. Fig. visualizes the averaged computation time vs. prediction accuracy over 10 independent runs, and we can see that PI-BSNet obtained the lowest prediction error as well as the lowest training time. This is because PI-BSNet has more compact representation with B-spline basis functions, achieves zero initial and boundary condition losses at the very beginning of the training. In addition, thanks to the analytical calculation of gradients and Hessians, the training time of PI-BSNet is further shortened compared using finite difference, which is
12.5
Pi-DeepONet
10.0
PINN
7.5
HW-PINN
CAN PINN
5.0
2.5
Pi-85Net Pi-8SNet (FD)
10a
10a
Avg. Training Time (s)
Figure 4: Training time vs. prediction error
trade-offs.
When on the boundary of the C., the recovery probability horizon t = 0 is s(a, 0) but close the boundary with very small the recovery probahility is s(xr, 0) 0. With data the learning process goes unique solutions despite the presence of only one BC.
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Ground Truth
PI-BSNet
Absolute Difference
0.2
-
1
1
0.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.75
1.00
0.00
0.25
0.50
0.75
1.00
x
Figure 5: Test results on advection equations with unseen parameter.
given by PI-BSNet (FD). Additional ablations the effect of direct ICBC assignment can be found in
Appendix F.3
5.2 Advection Equations with Nonhomogeneous ICBCs
In this section, demonstrate the capability of the proposed P1-BSNet to handle nonhomogeneous ICBCs. We consider the following advection equation
as
Os +
0,
(19)
0
Or
where u E [0.5, 1.5] is a changing parameter. The domain of interest is set to be (1r, E [0, x [0, 2], and the initial condition is given by
s(z, 0) A sin (kr + aa)
(20)
where A = k = 2m, and a [0,27] is changing parameter. We train PI-BSNet with 3-layer fully connected neural networks with ReLU activation on varying parameters u E [0.5, 1.5] and a E [0, 2x), and test on randomly selected paramcters in the same domain. The B-spline basis of order 5 is used and the number of control points along x and t are set to be lg = 4 150. Note that more control points are used in this case study to represent the high frequency solution. Fig. visualizes the prediction results. The average relative L2 error across 30 test cases is 1.444 + 1.352 10
Additional visualizations for the parametric
results are shown in Fig. 13 in the Appendix.
5.3 Diffusion on Trapezoid
In this experiment, we demonstrate the capability of the proposed P1-BSNet to handle non-reclangular domains. Specifically, aim to estimate the probability that driftless Brownian motion in 2D (æ-y plane) with varying diffusion factor E [0, 1.5] along y direction, starting at point in trapezoid
Quarget = (zx y) E IR y E [0, 1}, x E [-1+ + 0.5 1 - 0.5
(21)
will exit the domain within given time horizon t [0, T] with T T= 1. Equivalently, want to compute the following value for all starting positions (x,w) E Qarret and & E [0,
s(x, v. t) = IP 3 t0,t] s.t. $ Ω (zo. yo) = (x, 3) E Starger)
(22)
We know that the exit probability s(x, y, the solution of the following diffusion equation
as 02s Øt 2
83 Oy
+
(23)
with ICBCs
s(0, 2, 0. v(Σ.y) E Starget,
(24)
st,,,,y) 1, WE TJ, V(x 8araggt
To solve this problem, we transform the target domain Quarget to rectangular mapped domain Qmapped
[(, v) E IR2 [0, 1), [0, 1]), and find the corresponding PDE. We train PI-BSNet with order d 3,
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0.1
(= 0.2
0.3
t= 0.5
t=1
Graund Truth
Pi-BSNet
Difference
100
Figure 6: Results on diffusion equation on the trapezoid over time.
number of control points lg = = and 4 100 on 10 uniformly sampled E [0, 1.5]. We then test the prediction results on unseen a. Fig. visualizes the results for one test case. It can be seen that PI-BSNet can accurately predict the diffusion evolution on the trapezoid with unseen parameter. The average prediction relative L2 error is 3.172 1.580 x 10-3 over 10 random testing trials. Domain transformation derivations and experiment details can be found in Appendix [D.] Additional visualizations for the parametric results are shown in Fig. 14 in the Appendix.
6 Discussions
In this section, discuss the practical considerations and limitations.
Choice of control point numbers. Theorem 4.3 suggests that higher number of control points provide a lower approximation error. In practice, when data are available, which is common for parametric PDE learning problems, one could tune the number of control points in prior to training to reach desired approximation error, or choose higher numbers of control points to ensure expressiveness of the model. general, smoother problems will require less control points reach a certain desired error tolerance, since B-splines are smooth functions. The number of control points has a direct impact on the training efficiency of PI-BSNet. Specifically, for n-dimensional problem, the number of neurons in the last layer of the coefficient. network scales (((") with & being the number of control points along each dimension. Empirically, the number of NN parameters and the training lime both increase the number of control points increases, as shown in Table $ in Appendix E.4 where we ablate different number of control points. In contrast, the test-time forward pass of the model remains computationally inexpensive and exhibits negligible dependence on the nunber of control points. In our implementation, and largely dominated by framework-level overhead (e.g., tensor allocation and dispatch PyTorch), rather than the spline evaluation neural network computation itself.
forward pass time is on the order of 10-4 seconds
Scalability to high-dimenssonal systems. For the number of points used to enforce the PDE loss, the proposed PI-BSNet, along with many PINN and neural operator methods, suffers from the curse of dimensionality. However, adaptive sampling techniques, such as those discussed in ŞZeng et al. 2022), can be effective reducing the mumber points required for training PINNs, especially when the PDE solution is rotcentrated certain regions of the domain. Similar ideas can be potentially incorporated into the proposed PI-BSNet framework through adaptive knot placement (Ych et al. 2020 which would reduce the number of control points needed and improve scalability.
For the number of network parameters, using fixed B-spline basis will result in exponential increase with the dimension for the proposed PI-BSNct. Additional treatment such as hicrarchical B-splines (Valentin) 2019).
control point adjustments (Yeh et al. 2020 Yang et al. 2004) and non-uniform B-spline representations (Pieg]
& Tiller 2012) can be potentially used to reduce the number control points, thus effectively reduce the network size. Neverthcless, the proposed PI-BSNet requires less control points than typical grid-basod optimization methods for PDE solving in such cases, thanks the representation capacity of B-spline basis functions.
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General domains and transformations. While the standard B-spline formulation primarily supports domains that are diffeomorphic to rectangulars, we show in Appendix C.2 that B-splines can effectively represent solutions on general domains, which motivates future work on physics-informed learning for such cases. In addition, coordinate transformations such as Mojgani et al. to effectively reduce the representation complexity, for idvection-dominal where dense control points are typically needed to accurately capture the coupled evolution between state and time.
2023 can be potentially loveraged
equations such as equation 19
Fixed basis vs. learned basis. With the use of fixed B-spline basis, the proposed method enjoys unique features such as exact enforcement of initial and boundary conditions, analytical derivative calculations for physics loss functions, inherent solution smoothness, and a compact, structured represenlation. These lead to practical advantages for systems like the one studied in Section [5.]] where we demonstrate faster convergence, higher prediction accuracy, and reduced computation time. the literature, there are also methods such as PI-DeepONet. (Goswami et al. 2023) that leverage learned basis. Correspondingly, longer training time is usually required due to the lack of structural advantages, with a potential gain in expressiveness. This distinction reflects the difference of architecture choice results in different capabilities and target applications. In particular, our method is especially suited for scenarios that demand strict IC/BC satisfaction, smooth solutions, and efficient training.
7 Conclusion
In this paper, we propose physics-informed deep B-spline networks (PI-BSNet), which incorporate B-spline functions into plysice-informed neural networks, to efficiently learn solutions of families of PDEs with varying ICBCs. With PI-BSNet, analytical derivatives are available for B-splines to calculate physics-informed losses, initial conditions and Dirichlet boundary conditions can be directly imposed through B-spline control points. We prove theoretical guarantees that PI-BSNets are universal approximators and have bounded generalization errors for elliptic and parabolic PDE families. We demonstrate in experiments that PI-BSNet achieves better training time and prediction accuracy trade-offs over various baselines, and is capable of addressing nonhomogeneous ICBCs and non-rectangular domains. Future work includes extensions to unparameterized domains and high-dimensional problems. Besides, the structured and smooth spline reprosentation underlying PI-BSNet provides a natural pathway to further extend the proposed framework, as fundamental B-spline properties such as convex hull guarantees, inherent smoothness, and noise-filtering behavior can support uncortainty quantification, robustness analysis, and other promising future directions toward reliable and
interpretable ohysics-infornee learning Prautzzch 2002
De Boor & De Boor 1978 Wahba 1990
Acknowledgments
This material is based upon work supported in part by the National Science Foundation under Grant number 2442948, and in part by grant from the Commonwealth of Pennsylvania, Department of Community and Economic Development. Saviz Mowlavi is supported solely by Mitsubishi Electric Research Laboratorics. We thank Jasmine Ratchford for insightful discussions that helped shape this work. We are also grateful to Giovanni Leoni from the Department of Mathematics at Carnegie Mellon University for valuable discussions on the continuity of PDE solutions.
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Yanjiao Wang, Shihua Tang, and Muqing Deng. Modeling nonlinear systems using the tensor network b-spline and the multi-innovation identification theory. International Journal of Robust and Nonlinear Control, 32 (13):7304 7318, 2022b.
Zhuoyuan Wang and Yorie Nakahira. A generalizable physics-informed learning framework for risk probability estimation. arXiv preprint arXiv:2305.00 06443, 2023.
Zhuoyuan Wang, Albert Chern, and Yorie Nakahira. Generalizable plysics-informed learning for stochastic safety-critical systems. IEEE Transactions on Automatic Control, 2025.
Musheng Wei. The perturbation of consistent least squares problems. Linear Algebra and its Applications, 112:231-244 1989.
Huaiping Yang, Wenping Wang, and Jiaguang Sun. Control point adjustment for b-spline curve approximation. Computer-Aided Design, 36(7):639 652, 2004.
Raine Yeh, Youssef SG Nashed, Tom Peterka, and Xavier Tricoche. Fast automatic knot placement method for accurate b-splinc curve fitting. Computer-aided design, 128:102905, 2020.
Ka Fai Cedric Yiu, Song Wang, Kok Lay Too, and Ah Chung Tsoi. Nonlincar system modeling via knot- optimizing b-spline networks. IEEE transactions on neural networks, 12(5):1013-1022, 2001.
Shaojie Zeng, Zong Zhang, and Qingsong Zou. Adaptive deep neural networks methods for high-dimensional partial differential ial equations. Journal of Computational Physics, 463:11 1232, 2022.
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Xuedong Zhu, Jianhua Liu, Xiaohui Ao, Sen He, Lei Tao, and Feng Gao. A best-fitting b-spline neural network approach to the prediction of advection diffusion physical fields with absorption and source terms. Entropy, 26(7):577, 2024.
advectionddiffusion
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Table of Contents
Introduction
1
Related Work
3
3
Proposed Method
4
Problem Formulation
4
3.2
B-Splines with Basis Functions
4
3.3
Pnfrrcc-Informed B-Spline Networks
5
T Theoretical Analysis
6
Universal Approximation
6
2
Generalization Error Bounds
Experiments
8
5.1
Recovery Probabilities)
5.2
Advection Equations with Nonhomogeneous ICBCs
Diffusion on Trapezoid
10
6 Discussions
11
Conclusion
12
Appendix
20
A Proof of Theorems
20
A.1 Universal Approximation (Theorem 4.3)
20
A.2 Generalization Error Bounds (Theorem 4.6
22
B Proof of Supporting Theorems and Lemmas
23
B. Proof of Theorem A.
23
B.2 Proof of Lemma A.2
24
B.3 Proof of Lemma A.3
25
Additional Results
27
C.1
Convex Hull Property of B-Splines
27
C.2
B-Spline Representation for General Domains
28
C.3
Generalization Error Verification
30
Other Boundary Condition Types
30
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D Experiment Details
31
D.1 Recovery Probabilities
31
D.2 Advection Equations
33
D.3 Dilfusion on Trapezoid
35
E
Ablation Experiments
36
E.1
B-spline Derivalives
................................
36
B.2
Optimality of Control Points
38
E.3
ICBC Assignment
39
Number of Control Points
40
8.5
Robustness and Loss Function Weights Ablations
E.6
Number of NN Layers and Parameters Ablation
40
F
Additional Experiments
40
1 Burgers' Equation
41
F.2 3D Heat Equation
41
A Proof of Theorems
A.1 Universal Approximation (Theorem 4.3
In this section, we prove that PI-BSNets are universal approximators families of PDEs of arbitrary
dimensions.
We first consider the one-dimensional function space L2((a, with La norm defined over the interval [a, b]. For two functions s, g E L2((a, b)), we define the inner product of these two functions
(s.g) = S' s(x)g" (x)dr,
(25)
where * denotes the conjugate complex. We say function s(x: is square-integrable if the following holds
(8,8) = / [s(x))drx < co.
(26)
We define the Lg norm between two functions s, g as
|| - gll2 = (L." |s(0)) -s()))))(x)) (g(x))))
(27)
We then state the following theorem that shows B-spline functions are universal approximators in the semse of L2 norms in one dimension.
Theorem A.1. Given a positive natural ramber and any d-time differentiable function s(x) E Le([a, b)), then for any e 0, there ezist a positive natural value & such that for all E > l, there erists realization of control points Ci, .., such that
||s
$112 sc
(28)
where
8(x) Σ c BB,((z)
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is the B-spline approzimation with Bx,a(z) being the B-spline basis functions defined in equation 6
Proof. (Theorem A.1 See Section B.1
Now that we have the crror bound of B-spline approximations in one dimension, we will extend the results arbitrary dimensions. We point out that the space L2( bl) is a Hilbert space (Balakrishnan) 2012). We
consider n Hilbert spaces L2( bil) for i = 1,,, n. We define the inner products of two n-ddiensional functions 9, E [2( bi] x ... x [Un, bn])
(5.9) = [.
bn
bi (,,)) (xi, ,n)dxr .. den,
(29)
and we say a function s: R + R is square-integrable if
hn b1
(8,8)
|s(21, (n)) ... drn < 0o.
(30)
Now we present the following lemma to bound the approximation error of n-dimensional B-splines.
Lemma A.2. Given set posilive natural numbers di,... ddn and d-time differentiable function s(21, 22, ... In) E L2([a1, b1] [a2,by] x... [n), Assume d 2 max/d1, dn h, then given any <> 0, there crist & E N+ of control points for each component i = =1, II..., n, such for all & 2 l, vi = n, there crists control points realization such that
||8 8-12 See,
(31)
where
fn
$(11,
,I))
Cha dn Be,d, (x1)
Bindd, (zn).
(32)
Proof- (Lemma A.2 See Section
B.2
On the other hand, we know that neural networks are universal approximators (Hornik et al., 1989 Leshno ct al., 1993), i.e., with large enough width or depth a neural network can approximate any function with arbitrary precision. We first show that given some basic assumptions on the solution of the physics problems, the optimal control points are continuous in the system and domain parameters and a, thus can be approximated by neural networks.
In this following lemma, we show that with Assumption 4.1 and Assumption 4.2 the optimal control points are continuous in terms of the system and ICBC parameters.
Lemma A.3. For any E N+ and two n n-ddmensional surfaces S2 Le([a1, bi] x (a2, b2] x x... x [an, bnl) being the solution of the physics problem defined in equation I with parameters an, un and a2 U2. Assume Assumption 4.1 and Assumplion 4.2 hold. Let Ci and C2 be the two control points lensors that reconstruct &1 and 82. For any e >> 0, 61,62 > O, there exist 81, 82 > such that ||an - a2ll < 88, and ||u - uzll < 82, and control points tensors C₁ and C2 with ||C1- C2ll 5(e) such that ||s| ll2 E1, ||s2 - -22ll2 < E2, and
||sı s2 <. Here 8(c) + O when 0.
Proof. (Lemma
A.3 See Section B.3
Lemma 3 shows that the optimal control points exist and are continuous in and thus can be approximated by neural networks with arbitrary precision given enough representation capability (Hornik
et al. 1989 We restate the universal approximation theorem for neural networks our context as follows, assuming the requirements for the neural network are met.
7The Borel space assumptions are since we consider Lg space which is Borel space.
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Theorem A.4. Assume Assumption 4.1 and 4.2 hold. Given any and a in a finite parameter set, and the corresponding R-spline approcimation control points tensor C= (c]e,x..xxx for the coefficient net Go(u, and Ve > O, when the network has enough width and depth, there is 0" such that
(G. S
(33)
Then, we combine Lemma A.2 and Theorem A.4 Lo prove the universal approximation property of PI-BSNet.
(Theorem 4.3
Proof- (Theorem 4.3 For any u and a, from Lemma
we know that there is f,, ln by taking Vi Vi = 1, 12), and the control points _ .x,]le S 61 for any 61 > 0, where equation 7 with the control points tensor
A.2
the upper bounds of control point numbers (by setting & 2
realization C := xc.,,x..xxx, such that [|$(21,02. #2, Fnn 8(21, 2(21,22,- 22,
is the B-spline approximation defined Theorem A.4 we know that there is a DBSN configuration Go(u, a) and corresponding parameters 0* such that ||Gg- [u. (u,a) - CII < e2 for any 62 0. Since B-spline functions are defined on bounded domains and are continuous and Lipschitz (Prautzsch 2002 Kunoth et al. 2018), and and are weighted sum of B-spline basis functions with weights Go-(1a and C (see equation 7), we know that | - $||2 S Lez for some positive constant L. Then by triangle inequality of the Ly norm, we have
Then, from
||
s||2 S || 6||2 + ||8 - a|12 Se Lez.
(34)
For any > 0 we can find ej and 62 such that c1 Lea to bound the norm.
A.2 Generalization Error Bounds (Theorem 4.6)
In this section, we show proof of Theorem 4.6 to bound the generalization error of PI-BSNet families of PDEs.
We start by giving a lemma on PI-BSNet generalization error for fixed elliptic and parabolic PDEs. From Section 4 we know that PI-BSNets are universal approximators of PDEs. From Section 3 we know that the physics loss Cp is imposed over the domain of interest. Given this, we have the following lemma on the generalization error of PI-BSNet for elliptic and parabolic PDEs with fixed parameters, adapted from (Wang
& Nakahira 2023 Theorem 6) and (Peng et al., 2020 Theorem 2.4).
Definition A.5. Let C IR" be a domain. We define the regularity of as
[B(z,r)
Ro inf reft.>00 min [10] mn/2 pm r(n/2+1)
(35)
where B(xrr) = {y OUER" [ly x|| < r) and I(-) is the Lebesgue measure of set (.).
Lemma A.6. WWang et al. 2025 Theorem 5) Assume the PDE is elliptic or parabolic, for any fined paramelers a and u, suppose that Ω E Tn is a bounded domain, E (δ) n C² is the solution to the PDE of interest where Q is the closure of Ω F(s, x) = 0, S defines the PDE, and B(s,z) = 0, x E Os is the boundary condition. Let Gg denote PI-BSNet parameterized by 0 and 5 the solution predicted by PI- PI-BSNet. If the following conditions holds:
(B +|]] < 81, where Uni is the wniform distribuution in Ub. 1. Ez-Uni(((,(o)) [IB(Go,
2. Ez-tni((((o)) [[F(G,, =) 62, where Unif( is the uniform distribution in Ω.
3. Go, F(Go, )), s are Lipschite continuous on Ω.
Then the error of 8 over Ω is bounded by
sup (3(x) s(x) s 8, + Md2 aE
(36)
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where M is constent depending S2 U and F, and
26 [26]
[11(2s] . -((=1))
81 max
21
Ra, [1b1
(Ra, 1T(--1//2 1)/2
(37)
28 [2] Ro|
62|[2] T(n/221 1)
82 = max
21-
IRa n///2
where R(:) is the regularity of (-), 1(-1 is the Lebesgue measure of set (-) and I is the Gamma function.
We then prove Theorem 4.6 to bound the generalization error for PI-BSNet on the family of PDEs.
Proof. (Theorem 4.6 The goal to prove equation 14 holds for any u E U and a A. Without loss of generality, we pick arbitrary U1 U and on A to evaluate the prediction error, and we denote the ground truth and PI-BSNet prediction as 81 and 81, respectively. From Assumption 4.5 we know that there are u2 E Uuain and 02 Auaan such that |lu1 u2ll2 S Au, || - a2lla Aa. Let $2 and &2 denote the ground truth and PI-BSNet prediction on the PDE with parameters and 0r-- Since the conditions in Theorem 4.6 hold for all u E Utrain and E Arrain and 81 and ó2 are taking the maximum among all a, we know the following inequality holds due to Lemma A.6
sup 2Ef(an)
(3((2) =2(2)) S 8, + Md2
(38)
where M is constant depending on Q(a2), 2x(22) and F, and 51, 82 are given by equation 15 Note that the domain considered is (a(1),, as eventually we will bound the error in this domain. Necessary mapping of the domain is applicd here and in the rest the proof when a1 + 02
Since and U are bounded, and from Assumption we know the PDE solution continuous in u and o, we know the solution is Lipschitz in u and a. Then we have
sup (x) - 82(x)] = ||st - 82/600 S Kills1 - 821l2 S In + Aa),
(39)
=Ef(a.)
where K2 and I, some finite constants
Evans 2022
Chapter and Chapter 7).
Lastly, from Assumption 4.4 we know that the learned control points from the cocfficient network Gø(u, a) are Lipschitz in u and a. Since the B-spline basis functions (,,((z) are bounded by construction, we know that
sup rEf((a)
- -z))))) = || - 82/000 S K2ll31 - 82ll2 S L2(Au + Da),
(40)
where K2 and Lg some finite constants.
Now, combining equation
38 equation 39 and equation 40 by triangular incquality we get
sup xE((an)
(z) - s1 (1x)
sup (x)( - 82(x)1 + sup xEf((a))
[81(z) - 82(x))
sup [51(x) (x)) x)l((a)
(41)
=Ef((a)
<8 M + i(Au + Ao).
where M a constant depending on parameter sets A, u, domain functions Q Do and the PDE F.
L Li + La Lipschitz constant.
B Proof of Supporting Theorems and Lemmas
B.1 Proof of Theorem A.1
Proof. (Theorem A.1
From Jia & Loi
1993 Strang & Fix 1971 we know that given d the least square spline
approximation of 8(x)
Σ C: Bi, be obtained by applying pre-filtering, sampling and post-filtering
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with Lg error bounded by
8||2 S Ca T ||s(00]
(42)
where Ca is a known constant (Blu & Unser and || the norm of the d-th derivative of s defined by
1999p, T is the sampling interval of the pre-filtered function,
1/2
1/00
2 2d ((w))
(43)
2T
and S(w) is the Fourier transform of s(x). Note that given s and d, ||s(d]| is a known constant.
Then, from Unser 1999) we know that the samples from the pre-filtered functions are exactly the control points e that minimize the L2 norm of the approximation error. In other words, the sampling time T and the number of control points é are coupled through the following relationship
T 1-1
(44)
since the domain is [a, b] and is divided into l equispaced intervals for control points. Then with q being the samples with interval T,, we can rewrite the error bound into
(4-9)²
a
d ||
|| 10-88125 Ca
(45)
Thus we know that for Ve > O, we can find i E N+ such that for all 12
|| - 8112 S (b Calls(c
Se
(46)
(E 1) 1)d
because for fixed d the numerator is constant, and the L2 norm bound converges to 0 / +
0
B.2 Proof of Lemma A.2
Proof. (Lemma A.2 For given .,,... Lns let C
== [cle xx.xxx.. be the control points tensor such that
'.. || Fn) 8(x1, ... . is minimized. Let (x',, ,aa) denote the knot points in the n-dimensional space, i.e. the equispaced grids where the control points are located. Then from Theorem A.1
and the scparability of the B-splines (Unser et al. 2002 Section IV-E), we know that
[" 0o
h1 (s 8)(s (21, +2 ... ,,,)d S EZ
(47)
where Ex (b-a) Ca, [1
This shows that the L2 norm along the æ1 direction at any knots points (I2, ... 2/h) is bounded. Now we show the following is bounded
(21 11dd
Jaz |m
b2 b. (s - 8) - (((s-8)*(ee (1,,,
I2, ... ",,,,, dd
(48)
We argue that s is Lipschitz as is defined on a bounded domain and is d-time differentiable, and also Lipschitz as B-spline functions of any order are Lipschitz (Prautzschi, 2002 Kunoth et al. (2088) and is finite. Then we know that (s- 5) =)) is Lipschitz with some Lipschitz constant La; along dimension for i= 1,2,....
Let {a) (o("))/. 15= denote the equispaced knot points on Ja b.] with spacing b-a; Define a partition {1rg)s of ê:-1
(a,, b.] such that for all x, E Lijj |xs - ا(موءم S he-as 15- Then for i = 2, for all x2 E 12,1
(ss (s 8)* I2, ... 22)) - (s -s((s 8)" (211 () Zg.... ... ,2,22 < Les 2( 1)
b2 - 02
(49)
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Then we have
b2
(s () - (21, 1. ( ddrdd
(50)
1/2
bb
( [s(s 8) (w) 22, uag. ... s das
11.
1=1
62
(s 8) (01, 2 ./.. ... I/a dirddr
j=1
12.
11
82
+
I(s - () (x1, 22 x2. .. ,²n) - (ss 8)(s 8)" (21, (0 ..s.... .. dd)
(51)
j=1
12. da
1/2
b1
b2 ph
Lxs b2 dr dridx2
8)(8 ()) 2.. ..'. ... drrddr +
Jn
2(12 1)
(52)
j=1
12.3
02
(b2 10-a0 ab) - a1)
$ (b2 - 02)
Ea Laz
621.211
(53)
2(62 2(2--1)
where equation 51 is the triangle inequality of norms, and equation 52 due to the Lipschitz-ness of the function.
Similarly we can show the bound when we integrate the next dimension
(s 8)" (0i, ra, a4. " sa)ddidozdss
(54)
52
01
Fg
b2
pbi
bb pbz pb)
10
bg a3
:2, 22, u3 A'.." ... "ar.,,","",aaaa (d)ddydddday (s - () 8) (x)
+
Lus 2(1s 1)
day de2drz (55)
1=1
1g.1
an
ag ooz J
(by (b2 02)(b1 -a1
(bs
aa)
fx + Tizg
(56)
2(13 -
We know that E21 bound E,,,". za that
+ O when l, + 00 for = 1,2,3. By keeping doing this, recursively we can find the
[/ ... 1
br (s 8) - (x), , rr,dxI
... dxn S Ezgon
(57)
Jay
where the left. hand side exactly || .2. Fn.) (nn)-8(1 - when l + for all = 1. n. Thus for any > O, we can find & for = 1,2, ,n such that for all
Fn 1 and the right hand side Cry,." 0
L, 2 l, there exists a control points realization such that
[1s(21,02,"" ,nn) $(21, "2, ,2,)ll2 Sc
(58)
B.3 Proof of Lemma A.3
Proof. (Lemma A.3 From Assumption 4.1 we know that there exists 8,, 0 such that ||a: a2l ò1,
and ||u] - uzll < 82, and ||s s2l12 <
Now we need to prove that there exist control points tensors C₁ and C2 with ||C C2 || < 8(e) such that ||s] 31 E1, ||s2 - 82 < 62. We prove by construction.
We first construct surrogate functions 8 and & by interpolation of $1 and 82, then find B-spline approximations $ and & of the surrogate functions. The relationships between 8, and visualized in Fig. 7
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B-spline Approximation
Interpolation
S
$
2
2
Figure 7: Relationships between ground truth s, interpolation s, and B-spline approximation used in the proof of Lemma A.3.
For the two surfaces 81 and 82, we first find two continuous functions 81 and 82 for approximation. Specifically, 51 and 82 are interpolations of sampled data on 81 and 82 with N; grids along i-th dimension, i = 2, n. Since Assumption 4.2 holds, from Lemma A.2 we know that there exist. E N+, E N+ and control points C1 and C2 of dimension l 22 ... en such that - e1/2, || - s2ll2 < e2/2. We also know that the optimal control points are obtained by solving the following least square (LS) problem to fit the sampled data on 81 and 82.
C1 in Bi (211) Big,d (12) (12) Binnd, (2rn) (211, ,x), Vo Di,
12=1
(59)
Σ
f2. dn Bi Bisds (12) ... Bin (Zn) = $2(x1,
,,,) VI E D2.
1=
where D1 and D2 are the sets of all sampled data on 81 and s2 and are assumed to be sufficiently dense on Q. Then, write the LS problem into the matrix form follow.
AiCi bi,
(60)
A₂C² b2,
where A, A2 and b2112 < 8'(e), since 81 and 82 are Lipschitz on the bounded domain with [|8] 1001-2l12 < Here (c) 0 as < Without loss of generality, we consider cases where the sampled data on s1 and 82 is dense enough such that both A and Ag are full rank. Then by results from the LS problems with perturbation (Wei 1989), we know that the difference of the LS solutions of the two problems in equation 60 is bounded by
|C C2ll < 5(c).
(61)
where Ō(E) +
Since 81 and 82 are continuous functions defined on bounded domain, we know that both functions are Lipschitz. We denote L the larger Lipschitz constants of the two functions along dimension = 1,2, ,, i.e., Va = [I1, ",,, x' = = .. ,I/] E R",
1s (x) s1('') | S L 211 L2br2 - x66l + ... + L 1/yl. [s2(z) =2(x')) s 1 2'1 + L2(22 - 1'yl +.. + L Lnl2n z'1l-
(62)
Then we know
||s| F1l12 5 Lb -a1)
L2(b2 02)
L, (bn aan)
+... +
N1
N₂
N₂
(63)
(b, - -a))
+ Lz(b2 - a2)
L, (bn - an)
|2 52112 <
Ni
N2
Nr
since 31 and 82 are the interpolations of sampled data on s₁ and 8q with N; grids along i-th dimension. We
know that || 51 - 0 and ||82 52/12 when N₂ for all i. Thus, we find N1, No, N,
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B-spline basis functions
B- B-spline curve reconstruction
1.0
stk)
Contro points
BL,00
Bu str)
Knot points
B spline reconstruction ;
at
0.6
1.0
0.4
0.5
0.0
0.0
0.5
1,0
1.5
2.0
2.5
3.0
1.0
2.0
3.0
Figure & Visualization B-spline basis functions (left) and B-spline reconstruction (right). Note that with the clamped knot points, (0) B3,3(3) 1, so that the reconstruction $(0) = C1 and 8(3) ce to enforce hard boundary satisfaction.
such that ||s -51l2 < < |s2 52/12 c2/2. Then by the triangle inequality we have
||s - 81ll2 S ||8 - 81 ll + ||8] 51l12 61 61/2 61/2 22 c1
(64)
||s $2112 I | 52112 + || $2ll2 c2/2 2 c2/ E2.
C Additional Results
C.1 Convex Hull Property of B-Splines
Considering a one-dimensional B-spline of the form as 8(x) = cBa(x), where a E [a, 6], we have
8 8 [a,abb + [.
(65)
where
min Ci.
max Cz.
This property is inherent to the Bernstein polynomials used to generate Béxier curves. Specifically, the Bézier curve is subtype of the B-spline, and it is also possible to transform Bézier curves into B-splines and vice versa (Prautzsch 2002).
This property also holds the multidimensional case when the B-spline represented by a tensor product of the B-spline basis functions in equation Prautzsch 2002):
3 (an, b1]
[any bn] [g
(66)
where
min
Ci G2
7
max
Cis fa
This property offers practical tool for verifying the reliability of the results produced by the trained learning scheme. In the case of learning recovery probabilities, the approximated solution should provide values between and 1. Since the number of control points is finite, a robust and reliable solution occurs all generated control points are within the range [0, 1), i.e.,
C 0 c= 1.
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C.2 B-Spline Representation for General Domains
In this section, we present. results using Besplines to represent surface in arbitrary domains. Without.
loss of generality, consider 2D cases. Specifically, we consider original domain (u, E D _ [0. (0, 1],
and surface defined on a parametric domain via the following transformation
r(u, v) = (r(u, U),
y(u, v) gu(u,
z(u, v) f(xlu, v), v(u,))).
Note that since ON and By arbitrary, (z. v) can lie in arbitrary domains of interest.
We approximate the surface by reconstructing æ, and using tensor-products B-spline defined on (u, v), and with slight abuse of notation, we use to denote the approximation:
su,, v) = Cig Bydl))) s(u, E R
where the control points Cij are given by
Cly =
E R.
and (u) and By,((v) are B-spline basis functions of degrees in the u and v directions, respectively, defined on clamped knot vectors as Section 3.2 The vectors Cij are the coutrol points in R With this representation, the surface components can be written as the following three separate B-spline functions
nee
(2
v)
y cg
Baa(())Baa(o)
Bla().
While the (u, parameters lie on regular meshgrid, the actual surface domain in the (z, v) space is shaped through the mapping functions Ir and Jy, and can also be approximated by B-splines.
To sample points on the surface, we can evaluate a meshgrid on D and compute:
Tkt 9 (uk: ve),
ykk 9y (ux, ve),
Zkt f(zre, Vke).
where t = ,,...,,,,,,,, and f = , are the indices of sampled points in the and v directions, respectively. Similarly, the control points are arranged on a meshgrid over D. Due to the clampod nature of the B-spline basis functions, the first and last basis functions in each direction satisfy:
Bi,a(0) 1, B 1,
B1,((0) 1, Bn..
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and all other basis functions vanish these boundary values. This property allows direct control over the surface boundary through the corresponding boundary control points. For example, we can write:
Rv
(u) u=
Sx
Ba,(),)))
(67)
Ry =
82(1)) 4=1
B (zd(v)) mud
(68)
j=1
(1( v==
ci Bi,a(u),
(69)
8r(u) v=1
Σ
mn (u)
(70)
The same onc-dimensional B-spline representation applies to the y and components. This demonstrates that subset of control points governs the shape and values of the surface along the boundary.
This method generalizes naturally to dimessndimensional problems, since each surface component is represented independently using B-spline functions.
In the following, we show an example of approximating a surface on an annulus with the methods above. We consider a parametric domain (u, v) E [o, 1] x [0, I], and define transformation to an annular domain in I with inner radius Tinner 1 and outer radius Touter The radial coordinate R(u) is defined as convex combination of squared radii
R(u)= v( u)r + urouter outer
The angular coordinate is set 0(v) = 2TV. The transformation to Cartesian coordinates is then given by
s(u, v) R(u) cos c2s(2
v(u, v) = R(u) sin (2TT).
This maps the unit square [0, 1]2 to annular region in (r, ys-ace..
We define a scalar height function z(r, y) over the annulus, depending on the transformed radial and angular coordinates
=(u, v) = 0.3 (..3sin(//Ru)) (000))))).
Hence, we can represent the surface z on (x, y) with the (u, v) coordinates as follows.
x(I v) yuu, v)
R(u) cos (2TT) R(u) sin (2TT) 0.3 sin (3)( (1001))))
=
2 2(u, v)
We then generate data on a 100 100 equispaced grid in the (u, v) domain, and fit B-spline surface of order using 30 control points along each direction. We use this grid structure independently learn B-spline surface for each of the x, y, and coordinates, allowing to represent the geometry of the annular surface. The boundary conditions are enforced by scparately computing the approximating B-splines in equation 67 for each component along the boundary. To determine the remaining control points, we solve least squares problem based on data points.
Fig. g visualizes the results. We can see that the B-splines are able to accurately reconstruct the surface on the annulus. Note that the transformation from (u, v) (a, y) could be arbitrary other than the annulus, suggesting the possibility of future work to learn PDE solutions in arbitrary domains.
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Ground Truth
Reconstruction
Error
-152
3
-0.2
1
1
11
1
Figure 9: B-spline reconstruction of a surface on an annulus.
Max. Prediction Eroo
10a
10
Total Training Loss
Figure 10: Maximum prediction error vs. total training loss for advection equations.
C.3 Generalization Error Verification
In this section, conduct experiments to verify the derived generalization bounds in Theorem 4.6 Specifically, consider the settings for the advection equation experiments in Section 5.2 We train PI-BSNet for 2000 epochs, sample results at every 50 epochs, and visualize the marimum prediction error over the PDE family (30 parametric PDE instances) vs. total training loss in Fig. 10 It can be seen that the maximum prediction error is generally linearly depedent on the total training loss, which matches the provided theoretical results.
C.4 Other Boundary Condition Types
In this section, we provide experiment results for parametric PDEs with Neumann type boundary conditions. Specifically, we consider the following parametric diffusion equation
828 Os (x. t) u Ot
(I,
O,
(71)
0x22
with IC
s(x, 0) = cos(TT)),
(72)
and the following Neumann BCs both sides
Øs
Øs
00 t) = 0,
dx (1, 0.
(73)
dr
The equation has the analytical solution as follows
s(x, 1) cos((Tar)) -os
(74)
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Table Training time and average test relative La error for the Neumann parametric diffusion problem.
Method
Training time (s) Avg. test relative L2 error
PINN
914.4
1.697 x 10-2
PI-DoepONet
1064.5
4.459 x 10-2
PI-BSNet
148.6
1.932 x 10-2
Algorithm Training Inforee-Informed Deep B-Spline Networks (PI-BSNet)
Given: Training data D = {(u, a, 8) Nurain
Input: B-spline order control points number & loss weights wp, wa, wh; epochs E Trained PI-BSNet parameters 0"
Output:
1:
Compute B-spline basis By and derivatives By Ba and higher derivatives if needed
2: for epoch to do
for cach sample (u, s) D do
Predict control points: Č Go(u,a)
Assign ICBCs:
č[0,:] + IC, end] + BC Specific control points assignment depending actual ICBCs
6:
Assemble B-spline approximation: $(x) = (o(u,a)(z) via equation
7:
Evaluate physics residual:
8:
0,5(x,) 82))(x) using Ba, B'd, By via equation 12
F(8, x, u) for all x NE?
10:
Compute losses:
11:
L, 1 EEEp 1F x. u)12
12:
La = DD EXED (s(x) - 3(x:)1
13:
needed: L. = M Exem 2(3,
14:
Update parameters:
15:
optimize 0 with loss L = wpLp + wahs + whbs
16:
return 0"
We consider E (0.1,.55 and generated training data and 10 testing data with uniformly randomly sampled u. We train the proposed PI-BSNet with number of control points l, le =, with B-spline order d = for 5000 epochs. The data loss, PDE loss, and ICBC loss weights are set 5, 1, respectively. Note that impose the Neumann BC loss function too since B-spline does not support direct assignment for derivatives. The cocfficient network is 3 layer fully connected NN with 128 hidden width at cach layer. We compare the proposed PI-BSNet with PINN and PI-DeepONet with similar architectures under the same setting. Fig. TT shows the prediction visualization for different PDE parameter u, for all three methods. The training time and average prediction L2 errors are reported in Table ! It can be seen that the proposed PI-BSNet achieves similar performance as basclines, but is much faster due to the compact representation.
Experiment Details
In this section, we provide details for the experiments in Section 5 The general procedures for training the proposed P1-BSNet is shown in Alg. [ In all statistical results, test trials were selected by uniformly random sampling from the corresponding paramcter spaces. All methods were evaluated the same test sets to ensure fair comparison.
D.1 Recovery Probabilities
Training Data: The convection diffusion PDE defined in equation 17 and equation 18 has analytical solution
s(x, 1) = /' (a-aa x)
((a - x) - ut)
/ 3
exp
dr,
(75)
2T
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Ground Truth
Prediction Machol
Abs Error
PINN
Pl-DeepONet
PI-BSNet
PI-DeepONet
PI-BSNet
PINN
PI-DeepONet
PI-BSNet
44
Figure 11: Test results on diffusion equations with Neumann boundary conditions.
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where is the parameter of the boundary of the set C. = (a R a), and u is the parameter in the
system dynamics dz, = u dl + duz. We use numerical integration to solve equation 75 to obtain ground truth
training data for the experiments. We uniformly sample 40 random u E [0, and [0, 4] for training data generation.
Network Configurations:
PI-BSNet (proposed): We use 3-layer fully connected neural nelworks with ReLU activation functions for the coefficient network. The number of neurons for each hidden layer is set be 64. We use Adam the optimizer.
PINN (Raissi et al. 2019: ; A 3-layer fully connected neural network with Tanh activations used approximate the solution directly. The PDE residuals are computed via automatic differentiation. Initial and boundary conditions losses are imposed.
EPINN (Wang al. 2023 A 3-layer fully connected network with Tanh activation is used. The exact solution form is preserved via the function template + Lo. where So is approxinated by the work The spatial-temporal encoding is based on domain -ware transformations using o, = e-x_ and or = $.
a-2min
SFHCPINN (Li el al. 2024a Extends EPINN by applying sinusoidal leature embeddings (SIREN-style) to the imput coordinates (z, t) before passing them into the network. These features are linearly transformed and concatenated with parameters (A, a) before being passed to a 2-layer MLP with ReLU activations. The template + fo is still enforced.
(Chen et al. hidden layer width is replaced by a wider fully connected MLP (two hidden layers 64 neurons each with Tanh activation). This model emphasizes expressive capacity within the hard-constraint formulation.
HWPINN
2023
The same formulation g + 0z0:Jo is used as in EPINN while the standard
VS-PINN Ko & Park 2025 J: A learnable input transformation applied to (x, t) via trainable scaling factors before the hard-constraineel template enforced. This adds a degree flexibility to adapt unknown domain warping or scale mismatch, while preserving the g + du form.
CAN-PINN (Chiu et al. ) : The hard-constrained formulation used along with a finiteddifeerence approximation to compute second-order spatial derivatives in the PDE residual. The model uses a standard MLP with two hidden Tanh layers (64 neurons each) and prodicts the residual via custom finite difference
steps.
PI-DeepONet (Goswami ct al. 2023
A DeepONet structure is used with separate branch and trunk neiworks, each comprising two Tanh layers of 64 neurons. The outputs of the two networks are fused through an inner product after a linear transformation, enforcing parameter conditioning and spatial-temports separation. Initial and boundary condition losses are imposed.
PI-FNO Li et al. 2024b): This model is implemented based on the Tensorized Fourier Neural Operator (TFNO2d) module, and it operates on a 4-channel input grid of shape [λ, a, t). The architecture includes spectral convolution layers with 12 Fourier modes in each direction and a hidden width of 32. PDE residuals are computed using automatic differentiation on the input grid, and ICBC losses are imposed.
Note that we exlend the original PINN framework (Raissi et al. 2019) by augmenting the net work inputs with the PDE and ICBC parameters u and a, enabling the model to learn mappings to solutions over families of parametrized PDEs. Similar extensions applied to other PINN variants used basclines to ensure fair comparison.
Full Visualization: Visualization of prediction results for all methods is shown in Fig. I2
D.2 Advection Equations
Training Data: The advection equation defined in equation 19 admits an analytical solution
s(x t) sin (2mr ((x wt) mod 1) + a),
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PI-BSNet (Proposed)
EPINN
VS-PINN
PI-DeepONet
PI-NNO
1.0
0.0
10
10
10
10
-10
-10
5
4
-4
t
4
0
4
-4
2
X
X
X
1.0
1.0
1.0
10
1.0
0.5
0.5
0.5
Ground Truth
0.0
10
10
10
-10
5
10
5
-10
-10
t
t
0
0
0.5
X
X
PI-BSNet - FD
PINN
SFHCPINN
HWPINN
CANPINN
110
5
1.0
1.0
05
0.5
0.0
0.0
0.0
10
10
10
10
10
-
2
10
2
-10
X
4
4
X
4
1.0
1.0
1.0
1.0
1.
0.5
0.5
0.0
00
10
10
10
5
10
0
0
t
0
2
X
Figure 12: Recovery probability visualizations. Predictions in first row and errors in second row.
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Ground Truth
PI-BSNet
Absolute Difference
2
2
2
0.2
-1
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.00
0.25
0.50
0.75
2
2
2
0.2
1
-1
0.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
1
0
0.0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
X
X
Figure 13:
Results on advection equation with different random parameters.
which is evaluated uniform spatio-temporal grid with 100 points in [0, and 100 points in / [0, 2]. For each of the 100 training samples, uniformly sample u E [0.5, 1.5] and [0,2
Network Configurations: We train PI-BSNet with 3-layer fully connected neural networks with ReLU activation on varying parameters w E [0.5, 1.5] and E [o, 2T), and test on randomly selected parameters in the same domain. The B-spline basis of order 5 is used and the number of control points along x and are set to be 4g = & = 150. Note that more control points are used in this case study to represent the high frequency solution.
Additional Visualizations: Additional visualizations for testing results with different random parameters are shown in Fig. 13
D.3 Diffusion Trapezoid
Domain Transformation: The exit probability s(x, f, defined in equation 22 the solution of the following diffusion equation
as = 2 dz
82
82
+ a 0y
(76)
Ōt
where a E [0,2] an unknown paramoter. And the ICBCs are
s(0, x. y) = O, V(x, v) E Qargets
(77)
s(1, 21, v) 1,
VI E [0 V(z Mataggt
We define the square mapped domain
Pmapped (1, v) E R E [0, E [0.
(78)
and we can find the mapping from the target domain to this mapped domain
1 + 0.5g
u) = T(x, =
2 y/
(79)
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which maps the left boundary -1 + 0.5y of Stareet to the left edge 0 of mmapped and the right boundary x 1 - 0.5y to the right edge 1, while preserving y. The inverse mapping is then given by
(a, v) T (u, v) (-1 0.5v (2 v)u, v).
(80)
Note that the mapped domain Qmapped can be readily handled by PI-BSNet. We then derive the transformed PDE on the mapped square domain Smapped to be
0s 1
0 A((,,)) v) Ou
Os B(u, v)
0s
0 -B((,, v) 0s
0s
)
Ôt 22 --
=
Ou
d0
+
du
Du
+ D(u, v)
0v
(81)
where
A(u,0) 1 a(u-).))*
B(u,v) = a00.5 u), D(u, v) a(2 -)).
(82)
2-
The corresponding ICBCs are
(u, 1, E TJ, v(u E 09 mapped:
(83)
s(u, v, = 0, V(u, v) E Imapped.
For efficient evaluation of the physics loss, we approximate equation cross-term-free PDE
81
with the following anisotropic but
Ös Ōt = 2 (2 - v)² Ou
1
0²s
d²s
+
a
(84)
Ouz
Training Data:
The ground truth data is gemerated by solving equation 76 on the trapezoid domain using the forward Euler method, with time step size At = 0.001 and spatial resolution NN)) = (21, 21).
Network Configurations: We generate 50 sample solutions of equation 76 with varying a uniformly sampled from [0, 1.5], and transform the solution from Saagget to Qmapped via equation 79 training data. We construct P1-BSNet with 3-layer neural network with ReLU activation functions and 64 hidden neurons each layer. The number of control points are l, ly 20 and l, 100. The order of B-spline is set to be 3. We train the coefficient network with Adam optimizer with 10-3 initial learning rate for 3000 epoch. Note that the physics loss enforces equation 84 on Qmapped, which is the domain for PI-BSNet training. We use wa = and wp = 0.001 as the loss weights for training. We use a smaller weight for physics loss since the physics model is approximate. We then test the prediction results on unseen randomly sampled from [0, 1.5].
Additional Visualizations: Additional visualizations for testing results with different random parameters are shown in Fig. 14
E
Ablation Experiments
E.1
B-spline Derivatives
In this section, we show that the analytical formula in equation 12 can produce fast and accurate calculation of B-spline derivatives. Fig. 15 shows the derivatives from B-spline analytical formula and finite difference for the 2D space [-10, 2] [o, 10] with the number of control point l = lg = 15. The control points are generated randomly the 2D space, and the derivatives are evaluated at mesh grids with N₁ N₂ 100. We see that the derivatives generated from B-spline formulas match well with the ones from finite difference, except for the boundary where finite difference is not accurate due to the lack of neighboring data points. Note that since there is no fitting involved, the B-spline derivatives are the ground truth values. This illustrates the advantages of using B-spline analytical derivatives over finite difference during physics-informn training.
We further consider a case where the surface is approximated by B-splines. Specifically, we consider the following 2D surface defined on [-1, x [0,1], 1],
s(1, σ)) = sin (2m..)) cos(TT) + 0.1 2 0.05 (x - -x30 + (t 0.6)2
(85)
0.05
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0
0.1
0.2
t= 0.3
r= 0.5
t= 1
Ground Truth
PI-BSNet
Difference
Ground Truth
Pi-BSNet
Difference
622
Figure 14: Results on diffusion equation on the trapezoid with different random parameters.
(B (-Spline)
Derivative (Finite Difference)
Difference (1st)
02
0.
02
-10
2nd Derivative (B-Spline]
2nd Derivative (Finite Difference)
Difference (2nd)
Figure 15: First and second derivatives from B-splines and finite difference.
where the ground truth gradients and Hessians can be obtained. We fit B-spline surface with the number of control point é = &2 20 and order 4. Fig. 16 shows the derivatives from B-spline analytical formula and finite difference and Table 2 shows the computation time and errors. The derivatives are evaluated at mesh grids with N, N₂ 100. We can see that the derivatives generated from B-spline formulas are general more accurate than finite difference, this case even if the B-spline surface is approximating the ground truth surface, and the finite difference uses the ground truth data value.
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Ground Truth
D-Spline
D-5pf
0-Splime
2000
00000
60
-M
-08
60
+6.
Ground Trutt
8-Sptines 5,
14
FO.s
8-Spfiness Emoo
F3.G Emror
2000
30000
Groune
5-Spline
FD
9-Splioe Frme
FD S..
2.0000
60
-28
-MT
64
-08 -65
00
-03
-21
M
-C4
by
Error
06
04
030042
68
030TT
00077
00008
00000
Figure 16: Derivatives from B-splines after fitting and finite difference.
Table 2: Computation time and relative T22 error of the derivatives.
Method
Time (8)
8xr
Set
Star
B-Spline
4.43 x 10-4
6.78 x 10
4.42 x 10 10-4 4
2.23 x 10-4
3.14 x 10-3
4.59 x 10
FD
7.22 x 10-4
1.17 x 14
1.83
10 10-3
7.53 x 10
9.15 x 10
1.94 x 10-3
E.2
Optimality of Control Points
In this section, show that the learned control points of PI-BSNet are near-optimal the L2 norm sense. For the recovery probability problem considered in section 5.1
investigate the for fixed set. of
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Table 3: Effect ICBC enforcement strategies for PI-BSNet.
Method
Relative L2 error (mean + std)
PI-BSNet (direct ICBC assignment)
1.45 + 0.21 10-2
PI-BSNet (ICBC loss)
5.52 + 0.55 x 10-2
system and TCBC parameters = 1.5 and a = 2. We use the number of control points & = f2 25 on the domain [-10, x [0, 10), and obtain the optimal control points C* in the L2 norm sense by solving least square problem DDngg & Lin 2014) with the ground truth data. We then compare the learned control points C' witt O and the results are visualized in Fig. 17 We can see that the learned control points are very close to the optimal control points, which validates the efficacy of PI-BSNet. The only region where the difference is relatively large is near C25,0, where the solution not continuous and hard to characterize with this number of control points.
Control Points (Optimal)
Control Points (PI-8SNet)
Difference
2-
98
24
24
42
20
27-
26
20-
10
10
15
Figure 17: Control points.
E.3 ICBC Assignment
In this section, we investigate the effect of direct ICBC assigmment versus TCBC loss enforcement for the proposed PI-BSNet, and verify that explicitly assigning ICBC values through the B-spline formulation indeed benefits PDE solution learning. Specifically, we follow the experimental setting in Section 5.1 and train two variants: (i) the original PI-BSNet with direct ICBC assignment, and (ii) PI-BSNet without direct ICBC assignment, where ICBCs are imposed via an additional loss term wyLa as in equation 11 For both methods, the physics loss weight and data loss weight are to wp and W2 3, respectively, while the ICBC-loss variant uses w, 3. We repeat each setting 10 times and report the relative L2 error in Table The results show that introducing an additional ICBC loss degrades prediction accuracy and leads to larger errors. A reprosentative prediction case is visualized in Fig. 18 Although enforcing ICBCs via an auxiliary loss with careful luning can polentially yield reasonable training performance, directly assigning ICBCs is generally preferred, as balancing multiple loss terms is nontrivial and often sensitive to hyperparameter choices Wang
et al. (021aa;; Krishnapriyan et al. (2021).
Ground Truth
Direct Assignent
Error
Loss Function
Error
1.0
0.5
0.5
0.5
0.5
0.5
0.0
0.0
0.0
0.0
10
-10
10
5
10
5
-10
0
t
t
-4
X
Figure 18: Results on the convection diffusion equation with variants of PI-BSNet for ICBC enforcement.
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E.4 Number of Control Points
In this section, we invesligate the elfect of the number of control points the performance of PI-BSNet. The setting is described in section 5.1] Table shows the approximation error and training time of PI-BSNet with different numbers of control points along cach dimension. We can sec that the training time increases as the number of control points increases, and the approximation error decreases, which matches with Theorem 4.3 which indicates more control points can result in less approximation error. The test-time forward pass of the model remains computationally inexpensive and exhibits negligible dependence on the number of control points.
shows
Table 4: PI-BSNet prediction MSE with different numbers of control points along each dimension.
Number Control Points
10
15
20
Number of NN Parameters
4417
5392
9617
17092
27817
41792
Training Time (s)
241.76
223.53
247.39
295.67
310.83
370.48
Forward Pass Time (x10-* 8)
3.85
2.40
2.06
2.37
2.28
3.56
Prediction MSE (x 4)
5357.9
7.327
7.313
5.817
4.490
3.064
E.5 Robustness and Loss Function Weights Ablations
In this section, we provide ablation experiments of the proposed PI-BSNet with different loss function configurations, and examine its robust again noise. The setting described in section 5.1 We first. train with noiseless data and vary the data loss weight wg. Table shows the average MSE and its standard deviation over 10 independent runs. We can see that with more weights on the data loss, the prediction MSE reduces as noiseless data help with PI-BSNet to learn the ground Iruth solution. We then train with injected additive zero-mean Gaussian noise with standard deviation 0.05 and vary the physics loss weight wp- Table B shows the results. can be seen that increasing physics loss weights help PI-BSNet to learn the correct neighboring relationships despite noisy training data, which reduces prediction MSE. In general, the weight choices should depend the quality of the data, the training configurations (e.g., learning rates, optimizer, neural network architecture).
Table 5: PI-BSNet prediction MSE (noiseless data).
Wd
2
3
1
Prediction MSE (x10-))
36.76 = 12.16 12.91 = 10.40
10.21 # 3.99
9.28 4 6.78 3.95
1.36
Table 6: PI-BSNet prediction MSE (additive Gaussian noise dataa).
Up
4
Prediction MSE (x10 ?) 31.58 + 6.46 33.15
7.77 13.37
11.74
7.95 + 6.24
3.86
2.05
E.6 Number of NN Layers and Parameters Ablation
In this section, we show ablation results on the number of neural network (NN) layers and parameters. We follow the experiment settings in section 5.1 and train the proposed PI-BSNet with different numbers of hidden layers, each with 10 independent runs. The number of NN parameters, the prediction MSE and its standard deviation are shown in Table 7 We can see that with 3 layers the network achieves the lowest prediction errors, while the number of layers does not have huge influence on the overall performance.
F Additional Experiments
In this section, provide additional experiment results.
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Table 7: PI-BSNet prediction MSE with different numbers of NN layers.
Number of Hidden Layers
2
5
Number of NN parameters
37632
41792
45952
50112
Prediction MSE (x10-*)
1.12 + 0.43
0.90 + 0.42
3.17 + 2.46
3.12 +: 2.81
F.1 Burgers' Equation
We conduct additional experiments on the following Burgers' equation, by adapting the benchmark problems Takamoto et al. 2022 to account for varying system and ICBC parameters.
in PDEBench
Ōs as
Gs dx²
+ us
v
= 0,
(86)
dr
where v = 0.01 and u E (0.5,1.5) is a changing parameter. The domain of interest is set to be (x,t)
E
[0, 10) x [0, 8), and the initial condition
s(x, 0) =
(87)
where a 4] is a changing parameter. We train PI-BSNet with 3-layer fully connected neural networks with ReLU activation on varying parameters u E [0.5, 1.5) and [2, 4), and test randomly selected parameters in the same domain. The B-spline basis of order 4 is used and the number of control points along and f are set to be lg = 4 100. Note that more control points are used in this case study compared to the convection diffusion equation in section 5.1 as the solution of the Burgers' equation has higher frequency along the ridge which requires finer control points to represent. Adaptive basis functions can be potentially used to further reduce errors under the same number of control points. Fig, 19 visualizes the prediction results on several random parameter settings. The average relative L2 error across 20 test cases 7.294 + 1.908 10-2.
F.2
3D Heat Equation
We consider the 3D heat equation given by
a scr D
82
Or s s(x,
(88)
Ot
where == 0.1 is the constant diffusion coefficient. Here x = I2, I2 Fz} E RR3 are the states, and the domains of interest are Qry = G2y = Ozg = [o, 1], and Ω = [0, 1). All lengths are in centimeters (cm) and the time is in seconds (s). In this experiment we solve equation 88 with random linear initial conditions:
sxx, t 0) a1 I1 + 02 . 322 + 03 . 33 + 000
(89)
where 01,02, 0g [-0.5, 0.5) and og E [0, 1] are randomly chosen. We impose the following Dirichlet and Neumann boundary conditions:
s2x f/3 0) s(x, f/13 1) = 1,
(90)
a dx
à
s(z, 1| = 0) = 0x
s(z, f|. = 1) = 0,
s zz, f| = 0) =
Dz s(x, 2| 1) = 0
(91)
azz
We set the B-splines to have the same mumber l = 15 of equispaced control points in each direction including time. We sample the solution of the heat equation at 21 equally spaced locations in each dimension. Thus, cach time step consists of 153 = 3375 control points and each sample returns 154 = 50625 control points total. The inputs to our neural network are the values of a from which it learns the control points, and subsequently the initial condition surface via direct supervised learning. This is followed by learning the control points associated with later times, > 0) via the PI-BSNet method. Because of the natural time evolution component of this problem, we use network with residual connections and sequentially learn each time step. The neural network has size of about 5 x 10a learnable parameters. We also train standard
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Published in Transactions on Machine Learning Research (03/2026)
Ground
PI-BSNet
3.384
0.875
0.875
3.336
0.75C
0.025
-4
0,500
0.50C
0.375
3.144
0.25C
2.096
0.125
0.125
2,048
0.000
000
Ground Thuth
FI-BSNet
Difference
3.000
3.32
0.875
0.875
3.20
0.750
3.24
3.20
0.500
3.16
0.375
0.375
3.12
0.250
2.08
0.00C
2.04
2.00
Ground Trush
BSNct
Difference
1.000
1.00C
3.32
3.28
0.625
3.24
0.825
0.50C
3.20
0.5c0
3.16
0.25C
0.12
6.125
3.08
0.000
-0.125
Figure 19: Results Burgers' equations with different random parameter settings.
PINN (Rassi et al. 2019h for comparison. The PINN consists of 4 hidden layers with 50 neurons in each layer, with Tanh as the activation functions.
Fig. 20 shows the learned heat equation and a slice of the residual in the xı-t plane. It can be seen that with PI-BSNet the value is diffussigg over time as intended. Although our initial condition does not adhere to the heat equation as estimated by the B-spline derivative, we quickly achieve low residual, while PINN fails do so. The average residuals during testing is 0.0121 for PINN and 0.0032 for PI-BSNet, which indicates the efficacy of the PI-BSNet method.
42

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Published in Transactions on Machine Learning Research (03/2026)
Learned Heat Equations
x; - plane
X2 -E plane
Xg - plane
Residuals
PI-BSNet:
6:78
6:88
565
035 0.5 0AX 357
<lc 03
C3 0/ 0M
Figure 20: The lcarned solutions (left) and the residuals (right) for the 3D heat equations.
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Reviews:
[{"review_id": "IdMuKPcdDx", "reviewer": "Reviewer_6L5p", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a new physics-informed machine learning model whose goal is to learn partial differential equations (PDEs) accurately with changing initial boundary conditions (ICBCs). The key parts of the proposed method is to use B-splines with control points on high-dimensional surface of interest. To deal with the boundary conditionals, a Dirichlet model is considered. The PS-BSNet model introduced is mainly characterised in Figure 1 and consists of two main elements, a fully connected NN (3 layers for example) that predicts control points and the B-spline approximation, whose gradients are tractable. The authors introduce two additional theoretical results, one focused on proving that the B-spline model for the context of physics-informed models, and another on the generalisation error guarantees. Finally, the empirical results show both the strenghts and weaknesses of the proposed approach against a good proportion of last-SOTA methods in the literature. Main proofs and theoretical results are somehow omitted of the main manuscript and kept in the Appendix.", "claims_and_evidence_yn": "Yes", "claims_explanation": "**Points of strength:**\n\nI do think the paper is in good shape during large parts of the submitted manuscript. Sincerely, I actually liked the way PI-BSNet is designed and how it is flexible enough to combine the NN-modelled control points, the B-spline model, and the different constraints. Its adaptability to different dimensionalities and both on the range of the surface and the order of the PDE, looks like a significant merit. The connection with the rest of PINN models seems to be well drawn, at least for me, and not being entirely familiar with that particular subcommunity of works, the paper is understandable and the decisions taken meaningful. \n\nThe use and introduction of the B-spline model is actually important here, and I also like how the authors exploit its properties on the computational side. That's very cool. However, I miss a bit of discussion and more details on that side, as well as on the other. But I will add more details on this later on.\n\nThe two theoretical results are also significant, although I miss a bit the purpose of introducing them in that way. Let's say that they are not well integrated in the flow of details and descriptions. I understand more the importance here of the first one for the universal approximation idea, and the second one confuses me somehow.\n\nThe empirical results are not extremely long, but they are precise. I would have loved to have the same ones, perhaps a bit more extended, not on SOTA methods but on PDE examples or other types of benchmarks common in the PINN context.\n\n**Points of weakness:**\n\nAll the strengths given and perceived from the PI-BSNet get some shadows from the way the three losses are combined in Eq. 11) with the temperature hyperparameters. From the very first moment this comes into play, details and discussion is missing. Being aware that such a combination is somehow the first decision that one can take (linear combination), I can also say this particular task can get into issues or lack of good fitting easily. Multimodal models have told us a lot in the past. How optimization and such combination of losses lead correctly the PI-BSNet to get a good solution is a mystery to me, and Figure 3 partially confirms that --- why the data loss dominates and outperforms while not the others?\n\nThe tractability of the B-splines is super interesting and important, as I said before. However, I miss the discussion on its scalability, the way it gets balanced with the NN for the control points, and how the computation of the backward pass scales with the number of control points, order of the PDE, and data points.\n\nLast but not least, the theoretical results are somewhat out of context to me, and even if they provide important insights to this piece of work, they move the point of view away of PDEs, types of PDEs, families, and examples. From my perspective, I would've prioritized communication more on the side of providing insights on the type of PDEs, dimensionality, practical challenges, and issues of other PINNs that this model solves... instead of the universal approximation and the generalization. This last one does not really solve my concerns about the overfitting chances of using Eq. 11 under certain scenarios of data scarcity that are not super discussed. While going to the Appendix for checking the proofs on the first theorem, I had significant doubts and concerns about the potential existence of other similar B-spline results on the universality of the approximation. Is it possible that similar results are already there? I say this mainly bc the theorem and proofs are more based on the B-splines model rather than on the general PI-BSNet.", "audience_yn": "Yes", "audience_explanation": "N/A", "requested_changes": "I would be glad to see a significant update on the following points, that have also been mentioned before in the previous sections:\n\n- Discussion and details on the B-spline tractability and scalability of the forward pass.\n- More examples and context of the particular PDEs that the work could focus on in the near future, and what are the main practical motivations that authors had for the design of PI-BSNet and the choice of solutions for the parabolic and exponential families.\n- Discussion and details on Eq. 11, its role in the empirical results of Figure 3 and just explaining weaknesses or strengths of such an approach.\n- Accommodating better the theoretical results in the manuscript, being better integrated and less being an appendix to the main thread of the work.\n- Larger introduction, perhaps more practical and aligned with PINNs such as the work is really valued in that direction in the near future.\n- Changes to have the reader better informed on the power of B-splines, what theoretical results are there already (for instance, on the universal approximation thing).", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "7rKrOWAeTA", "reviewer": "Reviewer_YT1J", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper examines the problem of approximating solutions to parametric PDEs, where parameters may influence the initial and boundary conditions, the differential operator, and even the geometry of the domain. To address this challenge, the authors propose Physics-Informed Deep B-Spline Networks (PI-BSNet), a framework that combines fixed B-spline basis functions with physics-informed learning to estimate the associated coefficients and approximate the PDE solution. The reported results demonstrate the potential of this model for solving parametric PDEs. The authors also provide theoretical guarantees showing that PI-DBNs act as universal approximators for certain classes of parametric PDEs and establish corresponding generalization-error bounds.\n\n# Strengths:\n* Representing solutions in a fixed B-spline basis and learning only the corresponding weights offers a more interpretable alternative to fully neural PDE solvers. \n* A notable advantage of the proposed approach is that the B-spline construction enables the exact enforcement of initial and Dirichlet boundary conditions, thereby eliminating the need for explicit boundary-loss terms in the training objective. \n* By embedding physics information into the B-spline basis, the method eliminates the need to compute higher-order neural network derivatives during training, then speeding up the training procedure and addressing a difficulty in PINNs and related physics-informed approaches. \n\n# Weaknesses:\n\nThe structure of the paper is generally well organized, but several issues reduce clarity:\n* Figure 1 is confusing, is it possible to make it more concise?,\n* In the theoretical section, the introduction of the inner product and square integrability in the main body of the paper feels out of place, as these concepts are not actively used in the main paper. \n* The use of norms is unclear, particularly in Assumption 4.4. Aren’t the functions $G_{\\theta}(u,\\alpha)$ vector-valued? \n* The parameter spaces, $\\mathcal{A}$ and $\\mathcal{U}$ would be better introduced earlier in the paper to improve clarity.  \n\nThe theoretical analysis depends on standard universal approximation results for neural networks. The contribution would be significantly strengthened by results that leverage the approximation properties and convergence rates of B-splines. \n* Can you clarify the source of the claim that the solution spaces for any elliptic and parabolic PDE have the stated regularity? A reference would be helpful.\n* Do you have a proof or a reference for the proof of Lemma A.6?\n* Can you use equation 47 from the appendix and verify numerically that you can obtain the expected convergence rate for some PDE using your model and standard B-splines? I believe this may be more informative about the perfomance of the model.\n* Is it possible to obtain figures showing that indeed the model realy follows the generalized error bounds for some PDE?\n\nAlthough the method is meant to solve parametric PDEs, the experiments mostly show single-instance solutions, so it is hard to tell how well it works across different parameter values.  \n* It is unclear why only 10 and 30 test trials were conducted or how these trials were selected. \n* A metric to measure the error should be chosen across the experiments, I found different metrics accross the experiments.\n* Only one experiment compares the method to baselines, and some of those baselines are not really designed for parametric PDEs. To make a fair comparison, the baselines should be trained for their best performance, rather than just using similar architectures, since the models are fundamentally different. \n * It is mentioned that Neumann and Robin conditions still need loss-penalty terms. However, the paper doesn’t include any experiments showing how well the method handles these boundary conditions compared to other approaches.", "claims_and_evidence_yn": "No", "claims_explanation": "The experimental section does not fully demonstrate the parametric performance of the model. It is unclear why only 10 and 30 test trials were conducted or how these trials were selected. Additionally, there is no clear rationale for which metrics should be reported, as three different metrics are presented without explanation. Only one experiment compares the method to baselines, and some of those baselines are not really designed for parametric PDEs. To make a fair comparison, the baselines should be trained for their best performance, rather than just using similar architectures, since the models are fundamentally different.  Moreover, there are some details to be clarified about the theoretical part.", "audience_yn": "Yes", "audience_explanation": "This model tries to address a problem that lies at the core of Uncertainty Quantification (inverse problems, PDE-constrained optimization, etc.), even though the use of B-spline representations is not new. The model deserves merit for combining a well-established mathematical framework with machine-learning techniques in a way that may alleviate some of the limitations of classical methods.", "requested_changes": "* Try to address the comments regarding the presentation of the paper.\n* Report how the model behaves when using more B-splines or more data, or include results related to the convergence rates of the model (eq 47).\n* Add an experiment demonstrating that the model follows the generalized error bounds for a representative PDE.\n* Standardize the metrics used for the test trials.\n* Compare the method against the strongest relevant baseline. PINNs are not designed for parametrized PDEs, so a more appropriate baseline should be included.\n* Add an experiment evaluating the performance of the model under Neumann or Robin boundary conditions.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "CLLcFKwIIy", "reviewer": "Reviewer_qsJy", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper “Physics-Informed Deep B-Spline Networks” proposes to approximate the solution to a parametric PDE with changing boundary conditions using B-Splines, which are predicted by a neural network given the values of the parameters and the specific initial condition (ICBC). The key practical contribution here lies in fixing the boundaries of the B-Splines to natively handle Dirichlet-Boundary conditions by fixing the values of the control points at the boundaries, which is especially suited for ICBCs. The dependable simplicity of this approach is a strength, but previous work already combined PINNs and Splines [1, 2, 3]. Additionally, a theoretical analysis is provided, which consists of two parts. The first part is concerned with a universal-approximation theorem, which is stated point-wise for specific instances of initial conditions and values for the parameters of the PDE and assumes a PDE with a continuous solution. It is limited in its details, as it states that for a suitable PDE with solution s, a specific instances of PDE-parameter u and boundary parameter a, there exists a neural network G that predicts control points $c = G(u,a)$ such that the corresponding spline $\\tilde{s}$ is epsilon-close to $s$. But it does not state that for some epsilon and some suitable PDE there exists a G that maps u and a to c for some domain of u and a such that the corresponding $\\tilde{s}$ is always epsilon-close to $s$. The second is a generalization bound between the predicted approximation $\\tilde{s}$ and the true solution. This is stated in terms of the max. distance to a training-datapoint (in terms of u and a), lipschitz constants of both the NN G and the PDE (wrt to parameters u and a) and observed fit on the training examples. It demonstrates that the approach is valid, which is a strength, but it is also a very general statement. Then the experiment section follows, with 3 experiments. This section is limited as only the first experiment compares against baselines, but no other Spline-based approaches are compared to and the effect of fixing the boundary conditions is not ablated. The other two experiments have no comparisons. This is a clear weakness, a more thorough comparison would increase the quality of the contribution. These problems also appear to be non-standard.\n\n[1] Doległo et. al, Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions\n\n[2] Wandel et. al, Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs\n\n[3] Zhu et. al,  A Best-Fitting B-Spline Neural Network Approach to the Prediction of Advection–Diffusion Physical Fields with Absorption and Source Terms", "claims_and_evidence_yn": "No", "claims_explanation": "The theoretical statements appear to be accurate, but in its current form the experimental section is limited in its evidence: no baselines involving splines are presented and the effect of the different choices (approximating s with splines, analytical calculation of gradients and Hessians using the splines, fixed boundaries) is not ablated. As b-splines have already been used[1], it is unclear which choice drives the differences to the other baseline. Also, most experiments do not have a thorough comparison. \n\n[1] Doległo et. al, Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions", "audience_yn": "Yes", "audience_explanation": "The setting of ICBC is interesting and b-splines are tried and tested concepts with a straightforward way to fix the function values at the boundaries. The theoretical contributions on their own are probably of limited interest, as they are both very general but also not surprising given that splines with enough pieces can approximate a bounded, continuous function arbitrarily well. It is not clear whether there is something to be learned by the theoretical contributions alone.", "requested_changes": "In general, I am open to improving the assessment of the paper given improved experiments.\n\nThe critical request for changes are:\n - a more thorough comparison to existing spline-based approaches in the related-work section\n - an ablation of the different choices (b-spline, fixed boundary and analytical gradients/Hessians), showing their impact on the performance while comparing it to baselines. The experiment in E.1 for the analytical gradients/hessians is not sufficient as there are no comparisons and quantitative results, therefore the impact of the choice can not be judged \n - adding spline-based baselines (like [1, 2])\n\nGeneral things to improve:\n - notation in theorem 4.6: the Expectations do not mention their random variable in the function the expectation is over. This is repeated in the appendix. \n - notation in lemma A.2 (appendix): Equation 33 can be improved by not mentioning $x_1, \\dots, x_n$ to make it clear that it is not point-wise at $x_1, \\dots, x_n$ (so a notation like in eq 29)\n - in the proof of theorem 4.3: it would improve the writing by mentioning over what the upper bounds of control point number is\n- some citations are imprecise, e.g. citing textbooks without mentioning which result is used (e.g. citation Agmon 2010 on page 21 or Pratt 2007 on page 22)\n - in equation 43: is sigma^2 here correct? it is not clear where it is coming from\n - in equation 53: I think derivation is a bit imprecise as we know that for every $x_2$ (fixed) we find $x_2’$ (knot) that is close, but then we can not choose $x_2’$ and assume it is that every $x_2$ is close. You would have to enumerate the knot-points\n - notation in eq 68: it looks like $\\hat{s}$ is an element of the cartesian product between two intervals but it is a function. Similarly 69\n\nSmall things to improve:\n - in equation 63 and the following conclusion $|b_1 - b_2| < \\delta’(\\epsilon)$ (followed from $||s_1 - s_2||_2 < \\epsilon$): It would be good to elaborate here. If I am following correctly this derives the closeness of the $s_1$ and $s_2$ evaluated on a grid from the general $l_2$ similarity of the functions, which should be due to the lipschitz properties of $s_1$ and $s_2$. As this doesn’t hold in general it would be good to write that\n\n[1] Doległo et. al, Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions\n\n[2] Wandel et. al, Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs", "broader_impact_concerns": "no impact concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)