Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems
Authors: Sung Woong Cho
Hwijae Son
Venue: ICLR 2025 Poster (2025)
Keywords: Inverse Problems
Stability
Operator Learning
Physics-Informed Machine Learning
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
TL;DR: We propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without any labeled training data.

Abstract: Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without any labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Published as a conference paper at ICLR 2025
PHYSICS-INFORMED DEEP INVERSE OPERATOR NET- WORKS FOR SOLVING PDE INVERSE PROBLEMS
Sung Woong Cho
Hwijae Son"
Stochastic Analysis and Application Research Center Korea Advanced Institute of Science and Technology Daejeon, South Korea swcho95kr@kain ac.kr
Department of Mathematics Konkuk University Seoul, South Korea hwijaess konkuk ac. kr
ABSTRACT
Inverse problems involving partial differential equations (PDEs) can be seen as discovering mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typi- cally rely on large amounts of labeled training data, which is impractical for most real-world applications. Morcover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose novel architecture called Physics-Informed Deep Inverse Operator Net- works (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates estab- lished in the inverse problem literature to the operator learning framework, thereby providing robust theoretical foundation for our method. These estimates guar- antee that the proposed model, trained on finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.
INTRODUCTION
Deep learning has revolutionized numerous fields, from natural language processing to computer vision, due to its ability to model complex patterns in large datasets (LeCun et al. 2015). In the domain of scientific computing, deep learning offers promising approach to solving problems traditionally addressed by numerical methods, particularly when dealing with high-dimensional or nonlinear problems (Carleo et al., 2019 Karniadakis et al. 202I). However, direct applications of neural networks in scientific fields often encounter challenges such as the need for large datasets and difficulties in enforcing known physical laws within the learning process. These challenges have led to the development of specialized machine learning approaches that can integrate the governing principles scientific problems directly into the learning framework, improving both performance and generalizability.
Physics-Informet Machine Learning (PIML) has emerged as a powerful paradigm to address these challenges by embedding physical laws, typically expressed as partial differential equations (PDEs), into the structure of neural networks (Raissi et al. 2019; Sirignano & Spiliopoulos/ 2018). Rather than solely relying large labeled datasets, PIML incorporates the governing equations of physical systems into the learning process, allowing for data-efficient models that respect the underlying physics. This paradigm is particularly useful for scenarios where the amount of available data is limited, or where it is essential to maintain the consistency of predictions with known physical principles. PIML has been applied transfer, and electromagnetics, demonstrating the ability of neural networks to capture the behavior of complex systems while obcying their physical constraints (Mao al. 2020 Cai et 202 la
wide range of problems, including fluid dynamies, heat
Cuomo et al. 2022).
At the early stage, PIML methods focused on solving single problem instance, leading to growing need for real-time inference to multiple cases. Recent advances in operator learning have aimed to
Conresponning author.

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address this by learning mappings between function spaces, significantly reducing prediction costs. The Fourier Neural Operator (FNO) and convolutional neural operator use convolution operations regular grids (Li et al. kernel network, factorized Fourier neural operator, and geometry informed neural operator are de- signed to handle irregular grid structures (Li al.
to learn operators
2021; Raonic al. 2023), while the multipole graph
2020b TTan et al. 2023 Li al. 2023). To predict solutions at arbitrary points, Deep Operator Network (DeepONet) (Lu et al. (021) and Hy- PerDeepONet (Lee et al. 2023) employ neural networks represent output functions. Despite their efficiency, these methods require a labeled pair of functions for training. To address scenarios where such data unavailable, physics-infommed operator learning frameworks such as thysics-Informed Deep Operator Network (PI-DeepONet) and Physics-Informed Neural Operator (PINO) have been introduced (Li al. 2024 Wang et al.) 2021b), incorporating physical laws into the learning process to ensure accurate solutions even in the absence of labeled data.
Inverse problems, which aim to infer unknown parameters or functions from observed data, are a crucial area in scientific computing (Groetsch & Groetsch 1993 Tarantola| 2005). These problems often arise in medical imaging, geophysics, and structural engineering. While traditional methods for solving inverse problems rely optimization techniques or iterative solvers, recent advance- ments in deep learning have led to the development of neural network approaches for inverse prob-
lems (Aggarwal et al. 20188] Ongie et al., 2020 Fan & Ying 2020 Bar & Sochen
2021 Pokkunuru
et al. 2023 Son
Lee 20024).
Recently, various operator learning methods have been developed solve inverse problems by learn- ing the mapping between data and unknown quantities (Wang & Wang 2024, Kaltenbach et al. 2023 Li et al.) 2024), with Neural Inverse Operators (NIOs) standing out as a notable approach (Molinaro et al. 2023). However, being formulated in supervised manner, these methods require large amounts of labeled training data, which is often impractical in real-world applications, and they may overlook important physical constraints. Furthermore, while neural operators are highly flexible, they often lack the theoretical guarantees needed to ensure stable and accurate solutions, especially in ill- ill-posed inverse problems. This paper addresses the issues mentioned above by propos- ing a novel operator learning framework, termed Physics-Ifformed Informed Deep Inverse Operator Networks (PI-DIONs). We summarize our main contribution as follows:
We present PI-DIONs, a novel architecture that extends physics-informi operator learning for inverse problems. This approach enables real-time inference in arbitrary resolution without needing labeled data, effectively incorporating known physical principles.
By integrating the stability estimates of inverse problems into the operator learning frame- work, we provide a theoretically robust approach for learning solution operators in PDE- based inverse problems.
Extensive numerical validation demonstrates that PI-DIONs can accurately and efficiently learn solution operators across range of inverse problems while removing the dependence on large, labeled datasets.
2 PHYSICS-INFORMEN DEEP INVERSE OPERATOR NETWORKS
Let 52 C Rdd be bounded open set with a smooth boundary, and let 27 Ω x |0, T). Throughout this paper, we denote the domain by Ω and 2r depending whether the problem is time-dependent. We consider a generic PDE defined by: [u,, 0, x E Ω
B(u)= = O, x E 02 A large class of PDE-based inverse problems involves identifying an inverse operator,
p.m + s,
where Pm C 52 represents the measurement domain and s is an unknown quantity within the system of interest. Recently, research has focused directly parameterizing the inverse operator using ral networks F=l supervised by paired input-output dataset {()(() @ practical perspective, such labeled pairs are often difficult to obtain. Furthermore, this type of super- vised training tends to overlook important physical constraints unless explicitly addressed (Karn-)
((0) { However, from
adakis et al. 2021 Wang et al. 2021b). Existing operator learning frameworks do not parameterize
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and as functions; instead, they treat them solely as input and output under direct supervision, which prevents the incorporation of informed training. This challenge motivates the devel- opment of PI-DIONs, which propose in this paper.
PI-DIONs can be viewed as an extension of the physics-infommnt DeepONets consisting of three main components. The trunk network, usually modeled with Multi-Layer Perceptron (MLP), takes batched collocation points I as input and outputs a discretized basis functions (z(2) = (x),, 2 (1))(()),............))))))))) (x)). The reconstruction branch network processes the partial measure- ment data Pt... to produce the coefficients by (by,11, (a,,,,n,,....4. ...)) forming the complete pro- file of the solution (z) = by (o(x) Σh=1 (n,,d Similarly, the inverse branch network takes the same partial measurement data sm.. as input and produces the coefficients be (bc, bc,,,,,,,,...... ... bc.p) for the target function sc,0(x) = - Eth-ibe,lo..t be.nlo,n (x)). This parameteri- zation allows for the direct incorporation of the physics informmed training paradigm into inverse problems. as both u and sç.0 are expressed as functions æ.
The loss function C for training PI-DIONs is defined as the sum of two components, Lphysics and Latta as follows:
N.K Σ
N.M Σ
Lphasics
1
[N
[(u
NK
NM
k-1,==1
1=1,j=1 2,E0n
NL
Liata
(1) (xi) ui!
2
NL 1=
L C=\1phhysie + X2 Aotaa for some \1 and Xg.
where N denotes the number of samples, K, .M denote the number of collocation points in 12, OΩ, respectively, L represents the number of measurement data points in Qm, and u, represents the partial measurement 3i 2. Since model "1,00 8ç,0 functions of x;, the differential oper- ator N can be computed using automatic differentiation at the collocation points. Thus, the physics loss Ephysics ensures that both the solution and the target adhere the physical principles, while the data loss Ldata reduces the error between the reconstruction uy and the measurement data u l The overall architecture is illustrated in Figure [ We also present an initial version, PI-DIONs-vO, which features another natural architecture for solving inverse problems but demonstrates inferior performance, Appendix A
EXTENDING STABILITY ESTIMATES TO THE OPERATOR LEARNING FRAMEWORK
The theoretical foundation for the proposed PI-DIONs is threefold. First, we introduce the stability estimates for the inverse problems we considered. This stability estimate is a crucial component that connects the prediction error with the loss functions used for training PI-DIONs. Second, we demon- strate that this stability estimate can be extended to the operator learning framework demonstrating that small enough L implies small prediction error. Finally, we present universal approximation theorem for PI-DIONs, which guarantees that the loss function can be reduced to an arbitrarily small value.
3.1 STABILITY ESTIMATES FOR INVERSE PROBLEMS
We start with brief introduction to the stability estimates for inverse problems, beginning with an informal discussion of the connection between these stability estimates and the physics-informed loss functions.
Informal statements. Let u be the solution to equation | for a given s, with partial measurements taken over the subdomain Pm Ω. For any approximations u" and s*, the loss functions Cphysies and Latta are computed with N 1 (i.e., the single-sample case) at the collocation points { k=
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Reconstruction Branch Network
Measunements
Inverse Branch Network
(xa
L phystes + Edata
x
Trunk Network
Target funcsiona sçals)
Figure 1: Schematic illustration of PI-DIONs architecture. The reconstruction and inverse branch networks take partial measurement data as inputs and produce the coefficient vector for the solution and the target function, respectively. The trunk network takes the collocation point input and generates the corresponding basis functions for both the solution and the target function. All the networks are trained by simultaneously minimizing the loss function L Lphysics + Ldtta-
(15)) (1a)fer The stability estimate is given by the following:
||u - =|| + I|a" =|| S Ephysice + Ldata + E,
where e converges to zero as K, M, and tend to infinity.
This stability estimate ensures that the solution and the target can be accurately approximated in the L sense, given a sufficient number of collocation points and sufficient minimization the loss functions. Similar estimates for inverse problems have been explored in various studies, including those derived in works such as /Zhan et al. (202)) and /Zhang & Liu (2023a). We now present the formal statements for the benchmark problems considered.
Reaction-Diffusien equation.
As a benchmark problem, consider the reaction-diffusion equa-
tion:
O,u + Au = f(x)g(t), u(x, t) 0, (xx, 0) = (x)),
(I, E Ωr = Ω x [0,7],
(x, on x [o,T]. E
where Ω bounded domain. We assume that the function g(l) (0, is known in advance and satisfies g* 2 g(() 2 g 0 for some constants g and g We consider the final data u), ( along with the initial condition uo(x) and the Dirichlet boundary condition, such that Qm = 02r in this example. The objective is to approximate both the solution and the source term f using this boundary measurement. Let (u".f" denote the approximate solution to the inverse problem. By applying regularity theory and auxiliary function techniques, the stability estimates are given as follows (See Zhhng et al. (2023) for details).
||ʃ /1( + ||u- (llc(o.T):LL(0))
+ I CR(llapu Au" - ff'l +
xf (z, T)
+ [l((,,,) uz, ()r,)))l)a))) (IA(((,,))) ("r,))))
+1]((x, u w"(x, (x=, (000000000
where the constant CR depends on g+, .9' 9 and T.
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Helmholtz equation. We consider the Helmholtz equation in bounded, connected domain f2 C R with a smooth boundary aΩ. The boundary value problem is given by:
Nu (oVu)( + c(x)u(x) = f(x), x Q
Su
Bu dv (z) g(x), 02,
where 0 0o (a)) E (S2) and 0o < c(x) E C( for some constants do and co. Here, represents the outward normal direction on 0Ω. The inverse source problem involves determining the unknown source function f(x) by using the partial measurement of on a subdomain 92n-
Assume that a, c. and f are analytic within f2 and there exist constants and P2 such that
f²f(()) - Pio!
P2
for all multi-indices (O)) According to Zhang & Liu (2023a), there exist constants C1
and C2 (with C2 E (0, 1)) such that the following inequality holds:
( S Ci Cillulo..,
| 19 C210m//199) (2(m)))
where 2m and Pog are disjoint subsets of Ω. For the approximate solution (u", the stability estimates are given by:
C2 18ml u'
C2 18ml Rel
+ 11o du
Ca l8 gll La(0))
Ilf - ( s C(Ilu u
T + |IV (oVu)) -cu " f
dv
||u - 1w--" (1 & I (oVu") cu" + ( + Il -00 + l|o Ou
Ov gll(2(0n)--
On the other hand, the term | 100--"lla (l2,(..... can also be bounded by ||u u" 112 C2
as previ-
(R..).
ously established. By applying this bound, along with the assumption that || - ff fl (R. ))))) is less than E and using the triangle inequality, we conclude that there exists constant Cn such that the following inequality holds.
Il - 121)2)(0)) + llf - (a.)) + 1|ue u² (())))
Co fap
Co 18mml
+ llo Ou"
C2 10mll g
< CH - u²'l =0I112200 + V (oVn') - cu" + -*'l12(00)
g
dv
3.2 STABILITY ESTIMATES THE OPERATOR LEARNING FRAMEWORK
We introduce continuous version Edata of Ldata Im as follows:
Edata L. [un,0(x) u(x))] drdu(()),
where un is produced by the input u/, 2 and p(lt) represents the probability measure on the function space U. Similarly, let v(S) denote the probability measure the function space S which be induced by the inverse operator F-1 from (u(4). We assume the dataset ((ul), ( is sampled from p((f) and vSS) by using this probability measure.
We now state theorem that implies the difference between Edata and Latata becomes small when both the number of sampled functions N and the number grid points for Lauta are sufficiently large.
Theorem 1. Suppose that supueu [Iullz>(aa.) S R and aupn,e I1 (a.)) S R for some R > Consider an input-output dataset (((()) (( {())) generated through the following process. First, sample N functions {s()) y/ from the probability measure v(S) and compute the numerical solutions {u N 1i=1- Second, for all {ul03, sample the grid points for partial measurements from
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the uniform distribution 9am i.e., the grid points shared across all functions. If the number of functiios
functions N and the number of grid points satisfy:
1og(8Ne/0)
256R
12
log
2
log2
2²
then.
Edata S Edata +e.
holds with probability at least 1 1- 6, where No is constant depending on e.
The proof of this theorem leverages a symmetrization technique, analogous to the approach used in Baxter (2000). This method bounds the difference of Edata and Eatta by analyzing the discrepancy between instances of Ldota computed from different datasets. We can apply Hoeffding's s in- equality to prove that the difference converges to zero as the number of grid points and the number of sample functions increase. Further technical details of the proof are provided in Appendix D
A similar property can be derived for Lphysics, where arbitrary u, s sampled from μ(ll) and v(S) given sufficiently large number of grid points and samples.
satisfy the governing PDE, equation [ As before, we define the continuous version Ephysics of Lphysies as follows:
Ephysios Le [N(ug, (c,o())) dudu(()) [ [B(un, (x))] drdu(()),
where U and sc.0 are produced by the input u By applying a similar technique as above, we can prove the following theorem, which implies that Ephysics converges to Lphysics the number of grid points K, M and the number of samples N becomes sufficiently large.
Theorem 2. Consider the same sampling process for ( as described in Theorem holds. Suppose that supwEt
S Ry, supn,9 (lNu,,lll Los (Smm) S Ry, and supueu [ S RB, sup,,e 8 [lAu,, (l,, I Rg- Additionally, let and M denote the number of grid points for the interior Q and the boundary in, respectively, and shared across all 16,) If the number of sampled functions N and the number of grid points K, M satisfy:
(8log(16Nw/8) 8 8log(16Ne/5)
KK 64R log
64R log
N 2 max
log2
log2
22
then with probability least 1 8, the loss functions will satisfy
Ephysies S. Lepigsics + 6,
where Nx and Ng are constants depending c.
By applying Theorems and we can now derive the following theorem, which states that the output functions a and $2.0 PI-DIONs can approximate the true and with high probability over μ(l)) and v(S). Notably, previous results from Zhang et al. (2023); Zhang & Liu (2023a) established similar stability estimate, but they considered only the special case where the supports of μ((l) and p(S) consist of single element within U/ and S, respectively. In contrast, this paper aims to extend the result by considering more general distributions plll) and v(S). Consequently, accurately approximate and even with the partial
the theorem below implies that PI-DIONs measurement of unseen data u in U, which represents our main theoretical result for PI-DIONs.
Additionally, assume that the number of sampled functions N and the number of grid points L, K, M satisfy the conditions in Theorem 2 Then, for any u U, E S, and e > O, the following inequality holds with probability
Theorem 3. Suppose the same sampling process holds as Theorem Z
at least (1 26) - 2 Ephyyiie fatate)
Ve
I - - + - S VE.
Remark 1. Theorem 3 applies to the inverse source problem for the reaction-dee difffiniiif equation using boundary measurements, well as to the inverse source problem for the Helmholtz equa- tion using internal measurements. We believe that this result can be generalized to other inverse problems, where stability estimate for a single instance can be derived.

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Je) True
(b) Pertiel messuremeett
(C) Prediction taall,
(d) Target function
1.
2.0
The TXC
Peddrtion (eal))
02-
2.5
07
37
0.1
3.6
010
6.7
10 57
24
012
07
0.
1.0
Figure 2: A test sample and results for the inverse source problem of the reaction-diffusion equation. 30 30 rectangular grid. (b) Partial measurement data u anr collected from the true solution on the same grid. (c) Predicted solution U, on 200 200 rectangular grid. (d) The source function (ax) (blue) and the predicted source function $ç.0 (red).
(a) True solution
3.3 UNIVERSAL APPROXIMATION THEOREM FOR PI-DIONS
Finally, we demonstrate that PI-DIONs can achieve small values for both Ldttaa and Cpkyvice This result builds upon the universal approximation property of DeepONet, which asserts that DeepONet can approximate certain operators, including nonlinear mappings between two compact function spaces. Specifically, we claim that if DeepONet accurately approximates the solution and inverse operators, the physics-informed loss Lphysics will be small enough. The proof this claim relies on the standard triangle inequality, under the assumption that both the solution and inverse operators are Lipschitz continuous with respect to the measurements. By combining this with the results from Theorem ensure that PI-DIONs with a small value of C provide accurate approximations of both u and 8.
Proposition 1. (Universal approximation theorem for PI-DIONs). Assume that the branch network and trunk network in PI-DIONs have continuous, non-pon non-polynomial activation functions. For any given apput-oupput dataset {(u(*)
Q.. ,(())) N for any e >> 0 there exist appropriate parameters
S, and 0 for PI-DIONs such that
C = Lphysies Ldata S c.
4 EXPERIMENTS
We empirically evaluate the proposed PI-DIONs on three benchmark inverse problems of different
types. We present both the unsupervised PI-DIONs and supervised PI-DIONs, where the supervised (2) where
NK one trained with additional loss function L. + -i is (1) (xx)))
8
(i) Ik E Ω and & represents the label, so that the loss function becomes
= Ai Ephysiies + Az(Canta + L.), for some λ₁ and Az.
For the comparative analysis, we choose the supervised DeepONets and FNOs as baselines. The details of the training process, including the architecture and hyperparameters, are presented in Appendix B Additional experiments, including a sensitivity analysis, an ablation study, and a com- parison with PINNs, are provided in Appendix C The X-values selected based insights gained from training PINNs for each example. is worth noting that across all three experiments, we ob- serve that a larger Xa results in a smaller test error. This aligns with our intuition that, during the Xi
early stages of training, a large Ldata will push $ç.9 in the wrong direction, as uh,00 differs from the true solution, leading to inaccurate Lphysice. Finally, in Appendix C.3) provide detailed de- scription and additional experiments on integrating the variable-inpl operator network(Prasthoter et al. 2022) into PI-DIONs to address the case of irregular measurements (sensor points).

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(e) Portial meesurement
(b) Prediction teudl,
(ct Tue target furiction
(d) Prediction Sudes
05
28
10
00
677
2.6
08
1.0
000 07
P-
56
0.8
1.0
01
07
5/
06
08
010
Figure 3: A test sample and results for the Helmholtz equation. (a) Partial measurement v St., on an internal 40 x 40 grid. (b) Predicted solution o y) on 200 200 rectangular grid. (c) True source function f (x) on a 50 x 50 grid. (d) Predicted source function $ç,0 (x, y) on a 200 x 200 grid.
4.1 REACTION-DIFFUSIN EQUATION: INVERSE SOURCE PROBLEM USING BOUNDARY MEASUREMENT
We start by considering the reaction-diffusion equation
d.u + Au = F(x, t),
(x, t) E Ωr = Ω x [0,
u(2, = 0, u(z, 0) = (z)),
(x,t) 80 [0,
EEΩ,
where Ω [0. and 7 = 1. As it is hard to measure the internal source in many engineering applications, discovering unknown source function F(x, t) from the temperature distribution u(r, t) an important inverse problem. The inverse problem amounts to discovering the source function f()) from the initial data u(x, 0), final data u(2, T"), and boundary data (lanx(0,77) it well known f(x)g((t) is separable and g(t) is given in advance, then this
that if the source function F(x, problem attains a unique solution. Recently, /Zhng et al. (2023) proposed a physics-informed neural network for this problem, but it requires retraining when new data is introduced. In contrast, demonstrate that our method successfully learns the inverse operator, enabling real-time inference in arbitrary resolution without the need for retraining on the same problem.
We randomly sampled the initial condition un(x) and the unknown source function f(x) from the Gaussian random field. Using the central Finite Difference Method (FDM) on 30 30 grid, we computed the numerical solution and extracted the partial measurement dataset (x( 882 /i=1 pre-
cisely the boundary data in the space-time domain. Since the measurement data can treated as I-dimensional function, we used an MLP for both the reconstruction and inverse branch networks. As the solution and target functions have different domains, i.c., Uy : R² + R. and sç.0 IR R, we made reconstruction trunk network and inverse trunk network separately. The reconstruction branch, together with the reconstruction trunk network learns mapping from the partial measure- ment u aor to the complete solution profile 0r Simultancously, the inverse branch, also paired with the inverse trunk network, trained to approximate the inverse operator F w|oo2 +
We trained a PI-DION with 1000 unsupervised samples, where the data consists solely of par- tial measurements uI aaa We then compared the relative I errors against benchmark super- vised models, including DeepONets. Furthermore, we trained PI-DIONs with labeled training data ((u(1) I (()} to assess the performance of the unsupervised PI-DION. All models were evalu- ated on a test dataset of 1000 samples. As shown in Table the unsupervised PI-DION outperforms the supervised models and achieves a test error comparable to that of PI-DION trained with fully supervised samples. It is important to note that FNO is not applicable to this problem, as it requires the input and output functions to be defined on the same domain. In this problem, input function ul lan.. is defined on the boundary of 0h2r, while the source term ff is defined within the domain Q. This discrepancy between the domains requires training nontrivial mapping between these two dif- ferent spaces. We present test sample the true solution, partial measurement, predicted solution, true source function and the predicted source function in Figure?)
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(al Full messurement
(b) Prediction Meuds,
(ct Tuee target furittion
(d) Prediction Sudds,
1.
2.0
084
04
06
07 02
EX 1.0
00
07
P-5
56
0.8
1.0
07
6/ D6
22
Figure 4: test sample and results for the Darcy flow. (a) Full measurement v. on a 30 x 30 rectangular grid. (b) Predicted solution Un on 200 x 200 rectangular grid. (c) True permeability field on a 30 x 30 grid. (d) Predicted permeability field sç, on 200 x 200.
4.2 HELMHOLTZ EQUATION: INVERSE SOURCE PROBLEM USING INTERNAL MEASUREMENT
We consider the following Helmholtz equation:
-V (o(z)Vu(x))
c(z)u(x) f(z),
x S
(xx) Ou
g(x), E an,
Ov
where Ω [0, 1|2 and represents the outward normal direction on M. The objective of this inverse problem to reconstruct the source function f(z) for E St using internal measurement Am where 2m [0.2, 0.8]3. For simplicity, we set c = and = 0. This problem has been extensively studied from a Physies-Informed Neural Networks (PINNs) perspective in infzhng & Liu (2023b a). Here, we extend the problem into the operator learning framework and demonstrate
that PI-DION effectively solves the inverse problem by approximating the inverse operator without labeled data.
We again sampled f(x) from the Gaussian random field and computed the numerical solution using FDM, to obtain the internal measurement dataset. The reconstruction branch and the trunk network learn continuation mapping from u.
St... to u. Likewise, the inverse branch and the trunk network learn an inverse operator F-1; Qm ト Figure illustrates a test sample and results. Here, we employ CNN to model both branch networks and an MI.P to model the trunk network. We trained DeepONet and FNO with 50, 500, 1000 supervised samples, while PI-DIONs were trained with 1000 samples in both supervised and unsupervised settings. Table T shows that supervised PI-DION achieves the lowest test error, whereas the unsupervised PI-DION yields test error comparable to that the supervised FNO.
4.3
DARCY FLOW: UNKNOWN PERMEABILITY
Next, we consider the 2D steady state Darey flow equation:
- (a(x)) = f(x), x E Ω
u(x) 0, an.
In this example, u(x) represents the pressure field in a porous medium, defined by a positive perme- ability field (I)) The inverse problem involves determining the unknown permeability field (xx from the full measurement u(r). We randomly sampled the permeability from Gaussian random field followed by a min-max scaling, and computed the numerical solution u, using the fixed source term fzz y) 100x(1 z)y(1- y) and FDM 30 x 30 grid.
In this setup, the reconstruction branch, paired with the trunk network, simply learns the identity mapping from the measurement to the solution. In contrast, the inverse branch, along with the trunk network, learns the inverse operator F-1 u + a. Both branch networks are modeled using convolutional neural networks (CNN), while the trunk network is modeled using MLP. Given that the permeability field always positive, we impose a hard constraint on the inverse operator to ensure the output remains positive. We trained PI-DION with 1000 unsupervised samples, where

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Table 1: Relative I error of the predicted target function 8ç.0 computed over test dataset with the best model highlighted in bold. We trained supervised PI-DIONs with (2) (1, 100) and unsupervised PI-DIONs with (2) = (1, 100). Across all three benchmark problems, our PI-DION achieves the best performance, showing only small difference between supervised and unsupervised models.
Model
Train data
Reaction Diffusion
Darcy Flow
Helmholtz equation
DeepONet
50
33.60%
20. 15%
64.17%
w/ MLP, CNN branch
500
1.29%
9.06%
23.56%
(Supervised)
1000
1.10%
8.77%
7.86%
DeepONet
50
42.78%
28.79%
74.45%
w/ FNO branch
500
32.23%
12.78%
23.61%
(Supervised)
1000
21.10%
9.12%
11.30%
50
N/A
10.11%
38.42%
FNO (Supervised)
500
N/A
5.59%
28.23%
1000
N/A
3.41%
25.85%
PI-DION(Ours)
1000
1.04%
3.45%
5.64%
(Supervised)
PI-DION(Ours)
(Unsupervised) 1000
1.03%
8.10%
8.05%
the dataset consists of only. We compared the relative I.2 errors against the supervised models again. The results are summarized in [able|aand Figure 4 Supervised FNO demonstrates the best performance for this problem and the supervised PI-DION achieves a comparable test error. We believe there is room for improvement if stability estimate along with the corresponding physics- informed loss function can be further identified for this problem.
5 DISCUSSION
We explored inverse problems that can be framed through the identification of an inverse operator F between suitable function spaces. In this context, we proposed novel architecture called PI- DIONs, which eliminates the reliance on costly labeled data typically required for training machine learning models. By adopting a physics-informed approach, PI-DIONs directly parameterize the functions of interest-speee sprrestt-sseiffialllyy.. the solution 4y.0 and the target function 8ç.0- 55,0-thereby integrat- ing the underlying physical principles into the model. Additionally, we provided robust theoretical foundation for the proposed method by extending stability estimates for inverse problems to the operator learning framework. Our theoretical results demonstrate that PI-DIONs can accurately pre- dict both the solution u and the unknown quantity based on partial measurement data u nt.. This ability to work with incomplete data highlights the practicality and effectiveness of our approach. Additionally, we believe that empirically verifying the theoretical bounds on N, K, M, and will be an interesting direction for future work.
To substantiate our claims, we conducted series of numerical studies that showcased the superior performance of unsupervised PI-DIONs. Remarkably, these models achieved test errors comparable those of traditional supervised baselines, thereby indicating that unsupervised training can be as effective as its supervised counterpart in certain contexts. While the inference time of PI-DION is significantly faster than that of single PINN, it is important to note that the unsupervised training of PI-DION entails substantial computational costs (see (5) Therefore, accelerating the con- vergence PI-DION presents a promising avenue for future research, potentially enabling more efficient training processes and broader applicability to complex inverse problems. Furthermore, the integration of PI-DIONs with recent advancements in DeepONets, such as those presented in Prasthofer et al. (022) and Cho et al. (2024). will be interesting direction for future work. This exploration will not only refine our methodology but also contribute to the broader field physics- informed machine learning.
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REPRODUCIBILITY STATEMENT
We provide detailed experimental setups for each PDE in Appendix B Additionally, the proof' are included Appendix D and the source code submitted as supplementary material.
ACKNOWLEDGMENT
Sung Woong Cho supported by National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (KR) (RS-2024-000627555) and funded by the Korea government Hwijae Son was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 1TE-2022R1F1A10777777
(MSIT)
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A PI-DIONS-VO
Eppysics + bataa
Inverse Branch Network
fancters AgaOr)
Solation Mera(k)
Partal Mossurements
Forward Branch Network
Trunk Network
Figure 5: Schematic illustration of PI-DIONs-vO architecture. The inverse branch network takes partial measurement data input and produces the coefficient vector for the target function. The trunk network receives the collocation point z as input and generates the corresponding basis func- tions for the target function. The forward branch network takes the predicted target function 8c,8 as input and outputs the coefficients that form the solution U1,500. C. All the networks are trained by simultaneously minimizing the loss function Lphysics + Ldata-
In this section, we introduce an initial version of PI-DIONs, termed PI-DION-vO, which was consid- ered during the early stages of development. PI-DION-vO consists of the inverse branch network, the forward branch network, and the trunk network, similar to the final PI-DIONs architecture. How- ever, PI-DION-VO first produces the target function solution (ç.0(z) by combining the outputs of the inverse branch and the trunk network, then uses this prediction as input to the forward branch network to produce the final target function (z) The key difference lies in the input to the forvard(reconssruuctinn) branch: while PI-DIONs use the partial measurement, PI-DION-vO uses the predicted target function. The overall architecture is illustrated in Figure
PI-DION-vO also achieved promising relative test errors for the inverse source problem in the diittiooniiffioo equation. However, PI-DION consistently outperformed PI-DION-vO in both su- pervised and unsupervised settings (see Table Moreover, PI-DION proved be computationally more efficient than PI-DION-vO. Despite this, we believe it is worth discussing the architecture of PI-DION-vO in this paper, it represents
natural approach solving the inverse problem. The first part, consisting of the inverse branch and trunk network, can be viewed as an inverse oper- ator J-1, while the second part, the forward branch paired with the trunk network, operator F. Thus, PI-DION-vO can be interpreted as learning an identity operator I = F- F, mapping partial measurements the solution. We believe there are situations where PI-DION-VO outperforms, primarily due to its more natural architecture.
as forward
Table 2: Relative 2 test errors.
Train data
PI-DIONs-vO
PI-DIONs
Supervised w/ 1000
4.10%
1.04%
Unsupervised w/ 1000
5.62%
1.03%
B ARCHITECTURE AND TRAINING DETAILS
All experiments were conducted on a single RTX 3090 GPU, with the batch size determined based on available memory. For the three experiments, we used either 1,000 or 500 samples per batch. We used Adam optimizer for training, with learning rate le-3, We trained PI-DIONs and the baseline models for sufficient number of epochs until convergence achieved, as is typical with physics- informed training, which tends to have a slow yet reliable convergence process. We summarize the number of trainable parameters in Table
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Table 3: Number of trainable parameters for each model.
Model
Reaction Diffusion
Darcy Flow
Helmholtz equation
DeepONet
6K
70K
72K
DeepONet w/ FNO branch
45K
100K
150K
FNO
N/A
100K
150K
PI-DION
12K
100K
110K
B.1 REACTION-DIFFUSING EQUATION: INVERSE SOURCE PROBLEM USING BOUNDARY MEASUREMENT.
Both the reconstruction and inverse branch networks were simple Multi-Layer Perceptrons (MLP) each consisting of layers with 60 (input), 32, 32, and 32 (output) A ReLU activation was applied to each layer. Since the solution and target functions have different domains, i.c., U : R + R and $,,0 IR + R. designed separate trunk networks for reconstruction and inversion. Both trunk networks were also MLPs, with the reconstruction trunk consisting of layers with 2 (input), 32, 32, and 32 (output) nodes, and the inverse trunk consisting of layers with 1 (input), 32, 32, and 32 (output) nodes where a Tanh activation function was applied. The baseline DeepONet consists of the same MLP branch and MLP trunk networks and we did not consider the baseline FNO for this example.
The numerical solutions were computed on 30 30 grid, and the same grid used as input to the trunk networks for training PI-DIONs. The model was trained in full-batch setting, where the input to the branch networks has size of (1000, 60), the input the reconstruction trunk is (900, 2), and the input to the inverse trunk is (30, 1).
B.2 HELMIOOLTZ EQUATION: INVERSE SOURCE PROBLEM USING INTERNAL MEASUREMENT
For the Helmholtz equation, we employed convolutional neural network (CNN) composed of three Conv2d layers with 3 3 filters, each followed by the GeLU activation and max-pooling layers with a 3 x 3 kernel. Additionally, an MLP with 128 output nodes and GeLU activation was applied for both the reconstruction and inverse branch networks. The trunk network was an MLP with structure consisting of 2 (input), 128, 128, and 128 (output) nodes where Tanh activation was applied. The baseline DeepONet consists of the same CNN branch and MIP trunk networks and we adopted the implementation of FNO from Li et al. (020a)).
The numerical solutions were computed on 50 x 50 grid, and the internal 40 x 40 grid was used as input for the branch networks. PI-DION was trained with a mini-batch size of 500, meaning the input to the branch networks had a size of (500, 1600), and the input to the trunk network had size of (2500, 2).
B.3 DARCY FLOW: UNKNOWN PERMEABILITY
For the Darcy flow, we employed a CNN and MLP with the same architecture used for the Helmholtz equation. The trunk network an MI.P consisting of layers with (input), 128, 128, and 128 (output) nodes. To ensure the predicted permeability lies within the range [0, 1]. we applied a sigmoid activation function to the prediction sç,8. The baseline DeepONet consists the same CNN branch and MLP trunk networks and adopted the implementation of FNO from Li et al. 2020a).
The numerical solutions were computed 100 x 100 grid and then downsampled to 30 x 30 scale. PI-DION was trained with a mini-batch of size 500, meaning the input to the branch networks had size of (500, 900) and the input the trunk network had size of (900, 2).
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C ADDITIONAL EXPERIMENTS
In this section, we present additional experiments to further demonstrate the robustness and effec- tiveness of PI-DIONs.
C.I COMPARISON TO PHYSICS-INFORME NEURAL NETWORKS(PINNS)
We select the weights Å and X₂ for PI-DIONs based on insights gained from training PINNs. The inverse PINNs discussed in this section are unsupervised, meaning no supervision is provided for the target function 8. For each problem, we utilize two neural networks, each comprising three hidden layers with 64 neurons per layer, approximate the solution and the target function 8. The loss functions are defined similarly to those in PI-DIONs:
X Cphasics + 22 AoLata
All experiments were conducted on single RTX 3090 GPU. We used Adam optimizer for training, with learning rate 1e-3.
Table presents the relative L errors for different combinations of Interestingly, the unsuper- vised PI-DIONs achieve error levels comparable to those of single PINN, despite the significantly shorter inference time PI-DIONs compared to the longer training time required for PINNs (see
Table
Table 4: Relative L error of PINNs for different combinations of X.
(A1,
Reaction Diffusion
Darcy Flow
Helmholtz equation
(1, I)
0.90%
2.51%
8.44%
(1,100)
3.52%
1.47%
7.72%
(100,1)
7.55%
10.03%
11.49%
Table 5: Approximate training and inference time for PINNs and PI-DIONs.
Reaction Diffusion
Darcy Flow
Helmholtz equation
(1e+6 epochs)
(Ie+7 epochs)
(1e+7 epochs)
PINNs
Training
20m
3h
4h
Training
5h
24h
24h
PI-DIONs
Inference
2ms
5ms
5ms
C.2 SENSITIVITY ANALYSIS
The loss function comprising Lpkysics and Laata widely used in the Physics-Informed Machine (PIM))) literature. Additionally, numerous studies have introduced loss-balancing algo- rithms for these two components (c.g., (Wang et al. 2021a Son et al. (023)) ) by multiplying weights A and d2 to construct objective function:
L X1 Cphysics + Ca
Here, investigate how the performance of the proposed PI-DIONs can be further improved by incorporating such techniques. Specifically, we conduct a sensitivity analysis on the X-values. For this analysis, train seven PI-DION models with the following weight combinations:
b2) (100, 1), (10, 1), (1, 1), 0.1,, (0.1, 1), (1, 10), (1, 100).
The experiments conducted to solve the inverse
problem for the tion described in Section 4.1 Table presents the relative L errors for each model with different combinations of (Az, A2). Although the single PINN for the reaction-diffusion problem achieves
equa-
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Table 6: Relative I error for different combinations of X.
A2)
(100,1) (10,1)
(1,1)
(1,0.1)
(0.1,1)
(1,10)
(1,100)
Relative L crror 27.28%
14.84%
10.17%
13.81%
3.12%
4.79%
1.03%
the best error with (A1,A)) = (1, we observe that PI-DION achieves the best error when (A1, A2) 100).
We additionally conduct an ablation study to examine the effect of sample size on the reaction- diffusion equation described in Section [4. Table shows that the relative error decreases as the number of training samples (N) increases, eventually saturating around N 1000.
Table 7: Relative L error for different number of training samples.
N
100
500
1000
2000
Relative L error
28.73%
5.07%
1.03%
0.98%
C.3
VARIABLE-INPUT PI-DIONS
Reconstruction Branch
dda
seneor embedding
position embedding
Inverse Branch Network
x E DAL. irregular
L Epaystes Laata
Trunk Network
Target funclinssaaaaa)
Figure 6: Schematic illustration of variable-input PI-DIONs architecture.
Vanilla DeepONet requires that the sensor points (locations where the input function is sampled) remain consistent across all samples. However, the context of inverse problems, this constraint be problematic, sensor points may vary from sample to sample. To address this limitation, Prasthofer et al.] (2022) introduced variable-input deep operator networks, which allow for flexi- bility in sensor point locations. This variable-input architecture into our PI-DION framework. To evaluate this approach, we conducted experiments on a reaction- diffusion equation.
be seamlessly incorporated
generated a dataset ((n(*) s()) ) using fine computational grid Q_.
2y) 1/ considering the CFL condition De Moura Kubrusly (2013)). For each sam- ple (u(0) (.s())) (()), we randomly selected 30 collocation points in the spatial domain, resulting in D.. irregularly sampled data. Consequently, the set of collocation points varies across samples.
Although Prasthofer et al. (2022) originally proposed an attention-based mechanism for handling variable input, opted for simplified architecture. The overall structure is depicted in Figure 6 Both the sensor embedding and position embedding are implemented using simple multilayer perceptrons (MLPs). The final embedding is obtained by computing the inner product of their outputs. For the weights, we used (Y₁ A2) (1, 100). In this experiment, we obtained the L relative error of 3.83% which compatible with the original PI-DION results. This demonstrates that the proposed method can be effectively generalized to cases with irregular measurement points.
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D PROOF OF THEOREMS
D.1 PROOF OF THEOREM 1
Throughout this section, we assume the existence of sufficiently large constant It > 0 such that all functions u(x) U are contained within the range [-R, R[. Additionally, we assume that ue,,(z) is also contained within [-R, Rt for every 0, 7) and x E Q. This condition can be ensured by using bounded parameters 17 particularly when the activation function of the trunk network Lipschitz continuous. Specifically, u(az) is bounded (by the previous assumption on L)), and the input to the trunk network also be bounded, as we restrict our analysis to a bounded domain.
Next, we define the semi-continuous version of Ldata Ldata (17, 0, {xi follows:
Edota (11, 0) =
1/1
1m L
L
[nn)(x))= - du(2))
N
1m
(0) (2)) n (x)) dod(
L
1=1
where 70 denotes the PI-DIONs solution for the i-th component.
We first quantify the difference between Ldata and Ldata- Note that Edata (17, 0) is bounded by R where [22ml denotes the area of 2m Using this boundedness, along with Hoeffding's inequality, derive the following lemma.
Lemma 1. For any 0 and L 2 (ARR|9mm log 2/a², the following inequality holds.
(aitfe 9m, fu 1==1 Eu sup n,0
Idata (n, 0) Lduta 0, (x1 a
2P (mll 2m
fu Eu
sup Edaro (n. (m1)) Cdata (n, (2))()))
n,
Proof. We first focus determining a lower bound for the right-hand side of the inequality. Since the event on the left-hand side includes the event on the right-hand side, we can write.
Eallaanta P 2 E Pm 1 E Canta (n. O, (mr) - Ldata (n, 0, {x1)(=))
a
2 P ( E Sm, f E eul Lntetay, 0) Lanta (17, 0, f (21)(=1)
P
(m)tien E Im JN J Eu]/aaata
Ldata (n, 0, (21)1aa)
Now, we focus on bounding the last term the right-hand side using Hoeffding's inequality, lever- aging the boundedness of Ldata- Specifically, we have:
P (Lartia E Pms 1 E 2 [daata 0) Ldoca 0,
(x)) < 2
La
2 32R412mm
This follows from the fact that each term the left-hand side can be computed as:
Letara (1), 0) - Laata (17, 0, (x
N
1
1mml
1.
[ (6)
(x)))
u (xi))
/ [u]g(T)) u (x))2 dx dulu)),
11
N
nn..
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and
1mml L
4R S L 1mmm
(0) (1)) uu - (xx))
0 <
If L 2 (64 2 holds, we obtain the desired inequality.
We denote the function space that can be represented by (z))( as U. To quantify the complexity of this space, we define the covering number, which represents the number of elements in a set such that any arbitrary element can find sufficiently close representative in that set. This covering number will instrumental in deriving an upper bound for [danto- Ldatal.
Definition 1. For a given e > O, we define N(eū ú) as follows. Let Se = {(um 02), ... (une 9s be such that for any 0 U, there exists element un, Os in set Se such that the following inequality holds.
1m
(8) 92.00 (x1)
10 (xx)) - [Iu (x1) (x1)) Se
2L
Nee,U) U) is set to the minimum number of such S. Furthermore, we define Nele, U) to be the supremum of N(e,)) over any choice of (u()) N Fi=1 and {zili- Note that N₂ may be infinite.
Next introduce the subset II of the permutation group S2L. An element II is defined such that for all 1 < I < L. either (I) = l and a(l + L) = L, or o(l) = I + and o(l + L) =
With this definition, the following lemma holds.
Lemma 2. Suppose that No is the covering number for a/8 where S. = {um, ... (Unne: covering set for Ú. Then the following inequality holds:
P
(ail2
24 E 1m ful l sup n.0
Edata (17, 0, (21)7 Laara (n, (ailt a 2
o P
E
sup n,0
(Laata (n1 0, ( - Latta(1) (2o(T) 2
N₂
o II
Edata (7k. 0k: ( 17 - Cdota 8k (Ea()) |t,))=
Proof.
For a permutation and parameters 77, 0, assume the following inequality holds.
Ldara (1, 0, (Fonn - Ldata (n, 0, (uor)
We claim that there exists at least element in Se such that:
Ldata (Nk, Ox ( Lauta(Nk Ok, (a
Suppose, on the contrary, that no such element exists. By the definition of covering number, we can find a 0x, mh such that:
Pamm 2L
M (xz) - u.(xr))2 - [1,.0(()) - ui(zz))] a
2L
N
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Now, observe that the following inequality holds, where the last inequality is obtained by applying the standard triangle inequality.
24.
[un 0n (x1) u(ai)]² [un,(<r) u(xx)]²
2L
-=1
Ωm
N
-ul - (x)))) u (Fo(O)) Op
2L
2m
21
N
(1)
me ,z (Fo()) - u/ (xo()))² (1)o(())()))) - u0 (FFOC))
2L
N
1=++1
Lidara (n, 0, { Ldtata (n1, 0, ( (1)))))
Cdata (m,, Os, (1) )i=++1) - Laata (Mlt Os (1=())
Since the first term is bounded by a/8 and the last term is strictly greater than a:4 a/8 = ax8, we arrive a contradiction. Therefore, the claim holds.
Finally, we derive the upper bound for the result in the previous lemma using hoeffding's inequality with zero mean.
Lemma 3. For any n, 0, the following inequality holds.
a I (n, 0, (0o0))EL+1) -data (n1 0, 1, (zo
a) 4)
a²NL
4
s 2xxp
512/S2m
Proof. By definition of II, we observe that the condition:
Laara (n, 02 ( Lauta (n, 0, (1o())
can be written as:
oen Pm
[ (F u (=(0((41)]] - (xo0)))) a
This can be further transformed as:
1 1) o,
Im
Σ o. (((n),,(004))) u,(x44))) [un00((0)) (xxx))) a
=P =
for each
LN Σ
2'2
a2
< 2cxp (NNL,
512 12
where the last inequality is obtained by Hoeffding's s inequality. This holds because, by the bound- edness assumption on U, R. the following bound applies:
-AR² (1u - ( (100)) u0 (xi)33 4R²
Thus, the result follows.
Proposition Now if larger than 64R*|12,m log
log 2/a2, then the following inequality holds.
(mitta E 2mm ( E N
Eutata 0) - Laata (n, 0, (x1) (ar)(z)) >
sup
7.0
a
< <NNe
512(22,.122 RR
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For the proof, seek to find uniform bound for the following term.
22m n, Ju L
[
(un,((x))) u(xxi]²
[un,o(x) - -u(x)] ddz (d)))))))))
sup
5tm
We proceed similarly to the previous analysis and first establish the following lemma. We omit the proof here, as it can be shown similarly to previous ones.
Lemma 4. For fixed purameters y and 0, the following inequality holds if 64 log
a²
2m
L
IP
sup
[uno(xi) - u(xx)]²
(u))((x) u(xx]] doo dd =.)
L
9,0
Ju
Bm
Pm
S 2P
sup
u(xx)]
n.8
1u
L
1=1
L
1=1
where {11+L)ES H are sampled from Q using the uniform distribution.
Suppose that 1I defined as in the previous. We can now derive the following two lemmas using the same calculations as before.
Lemma 5, Suppose that Ng is covering number for g. Then, the following inequality holds.
P(o E II
sup n,6
/
()n.(()))))) - u(Ix(1))]
(,no(Fa((())) - u u( 2
Ju
L
L
i=1
No.
Qm
Unne 6n (Io()) u(<)<)))
Uma (Io(L+))) - u(
k=1
Lemma 6. For any E n. the the following inequality holds.
No
I.
L
[um or (Fo(L-))) - ul u(xo((1))]](d :
[une (Fo(1)) - u(xxcr))2
L
NLNL
< 2 exp
512R [mml
Finally, we can derive the following proposition, which shows that Edsta is close to Latta with high probability.
Proposition 3. T. is larger than 64Pt log 2/0², then the following inequality holds.
P (1i E Emm, (u (u Jil E EuN
Edara (n. 0) Ldata (11, (x1) > 20x
sup
n.0
NN 512/smm 12
I &NN.exp
Proof.
By triangle inequality, we have the following.
(m)) E Qm, (N E Eu IP
Edata Edata(1100) (1,,0)
On, (1))(=))) > 2x
sup n.8
< PP
(ailt E Ωm., fuN E v
sup Edata(n, Laotaan, (2: 1/ >
9.8
+P
( E Im, N E N
sup n.0
Edata (1, 0) Cduta (1, 0, [x1)=_1]11
By applying Lemma and along with Proposition
we can derive the desired result.
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D.2 PROOF OF THEOREM
2
Although Ephysice minimizes both two terms for the interior and boundary, the proof of Theorem follows similarly. Specifically, have:
S/d n
[N(un)()), sc,0(x)))] dxdp(e() / [B(((z))] dxdu(())
Ephysics
Jo
The only additional consideration is verifying boundedness of the derivative ((,,, (z)) and the definition the covering number. Once we redefine the covering number and assume that N(u, s) and N (u,,0 0= can derive the Theorem 2. Let S denote the set of functions comprising $ç,8-
$ç,0) are bounded by Ry, and B(u,, s) and ((u,,0) are bounded by Rg,
Definition 2. For given e O, we define Ne, 8) as follows. Suppose we find {(mm.8 B1 8m .ii); ..)..... (Uhn, Sn,.ca) such that for any uy,o E l and 8ç,8 E S there exists an element Um,.0p Opr Smx. Ob in set A such that the following inequality holds:
2K
ΣN>
Σ
[W me
(( Se,
in E Q
N
2K
N(, Ú) defined as the minimum number of such S. The covering number Nx defined as the supremum of S over any given ) and {ar)1=1- Similarly, NB is defined using the inequality:
N
2M
02 2M
[B (=)))² (x))) Se, 2y aΩ.
1=1
With the above covering number, we can derive bound on the difference between Lphysics and L.physies- Since the computation follows similar procedure, omit the detailed proof.
Proposition Now if and M are larger than 4 log 2/a² and 4R |012 log 2/a?, then the following inequality holds.
P
KK {ox
E EQ, (2511f E an, fu E U sup (physiico(1) Ephysics (n, fxx k (x513<1)1 4a
n,
a²NK
a²NM 32
< <8N.exp
+ SN.exp
The key part of the proof involves bounding the lefthand side of the inequality in the proposition. For both terms Sy Sa (z) sc,(()))] drdu(()) and fu Soq [B(u(z))) drdu(l)) the previous approach in the proof of Theorem can be applied to each part.
f E Q, (z11 E an, fN E u sup Ephysice (1, 0) Ephysico 0,, (151 4a P
n.0
< P
{un E EQ, JN E Eu
sup
C (11, 0) (a) ( > 2a
n.8
+
P
(r E on, 1N Eu
sup
Ephyetee (11,0) - Lphypsico 0, (1, { (ااحرر > 2x
n.8
D.3 PROOF OF THBBREM
3
We now provide the proof for Theorem which guarantees low prediction error for arbitrary func- tions and in the function space.
By applying Markov's inequality, we can ensure that samples (u, s) from the measure (μ(u), ((s)) approximately satisfy the governing equation, boundary condition and partial measurements consis-
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tency. Specifically, any for given O, the following probability bound holds:
sem P(u E / [uno(x)) u(xx)]dd 2x (N(un,())) dx + /
[B(u(z))] s a)
on
2 (Ephysics + Lda
Furthermore, with probability at least 1-26, we can bound the combined empirical losses Cphysics Liaro by the true losses Ephysics + Ldata as follows:
Ephysice + Esata S Lphystes + Ldata + 2e.
Recalling stability estimates for single element in LI we can conclude that, with probability at least 1 - (Cphysics + Ldata)/ a, the following error bound holds for E U.
||u,,0 u =P))) + ||5ç,0 a||2P(0) S
Finally, by choosing Ve, we obtain the desired probabilistic bounds on the solution error.
D.4 PROOF OF PROPOSITION 1
We first refer a relevant theorem which states DeepONet a universal approximator, allowing for close approximation with a sufficient number of parameters.
Theorem 4. (Chen & Chen (995)) Consider the case where X' is Banach space and K is a compact subset of X. Suppose that an operator G : 1 - S is continuous where U a compact subset of the infinite-din dinitt-ddinniinaal function space C(K;; IR) and S consists of the function whose domain is compact subset of Rd, Then for any e > O, there exists the unstacked DeepONet with the shallow branch net B and trunk net such that
(g(()(v) (B(u(z)),, (. <.
for all u E U and y y
While one might argue that the above theorem only applies to compact subsets of function spaces, making it insufficient for learning operators between diffnitedimmessionlll spaces, it is adequate for minimizing the discretized loss Cphysscc + Edata which is computed using finite samples and grid points. Moreover, DeepONet is universal approximator for continuous operator. In our case, we address an inverse problem that often satisfies stability estimate, ensuring continuity for both the solution operator and inverse operator. The detailed result is follows.
Proposition 5. (Universal Approximation Theorem for Operator). Assume that the branch network and trunk network in PI-DIONs have continuous, non- non-polynomial activation functions. Suppose that inverse problem has the following stability estimate.
1 - un (c + +| sıllra(0) I |u2la.. urlam 111200m))
For any given onput-output dataset {u((1 amm parameters T)s S, and for PI-DIONs such that
+00) and for any > 0 there exist appropriate
Ldata < c.
Proof. Consider the dataset {(u(() D. v(0) (00,)())) ( used for training. We aim to approximate the operator G u/sa.. su using PI-DION. Note that Gl(u()) ae,, Hy is continuous operator between (S2) and (1)(2 Furthermore, (1) (1p,.. J.V is a compact set, as it is finite. For any 1 E 0, by Theorem there exists PI-DIONs with 7,, Ç such that:
(G( (a. (x) s2,0)1 < ()) (2m (x) (u,, sc.0)l <, Vr Q Vi E (1,2, .N..
This implies that Lanta <
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The previous proof demonstrates that guarantee the reduction of Cdtta and therefore Edata- However, providing the reduction of Pphysica requires slightly different approach, relying on the universal approximation theorem for neural networks. The details are as follows.
Proof of proposition]
!
Consider the dataset {u(() u ())) used for training. We define three different neural net-
10m works By, Pc, To where B, (f (u (=)))=) and (x) ~ (0, ... 1, 1.... 0.0-... 0) and (0,.... ,0,00 1.... 0) R with the vectors in RRM having I in (i)-th and (i+ -) coor- dinate respectively. Meanwhile, the network approximates (r),.... uN (z), s(z), (x)).
By the universal approximation theorem for neural networks, for any multi-indes a, we have:
Σ |a|<=n
Oas (T()
(wi sI)), (x)))
If differential operator N is linear, is straightforward to show that:
Lphysics + Ldtta S
If A is nonlinear, involving product terms, the desired result can still be derived by applying the triangle inequality, assuming the boundedness of Thus, for both linear and nonlinear cases, the loss Cphssic + Ldtta can be made arbitrarily small, completing the proof.
24


Reviews:
[{"review_id": "eGsI9cTax0", "reviewer": "Reviewer_Y92e", "summary": "The authors propose Physics-Informed Deep Inverse Operator Networks (PI-DIONs), a novel architecture for solving PDE inverse problems without requiring labeled data. Theoretically, the authors extend stability estimates from traditional inverse problem theory to the operator learning setting, and prove universal approximation theorems for PI-DIONs. Empirically, the authors validate their proposed approach through experiments on reaction-diffusion equations, Helmholtz equations, and Darcy flow, achieving SOTA performance.", "questions": "- Questions about experimental results:\n  - What are the number of parameters of each of the models in Table 1?\n  - Could the authors provide a sensitivity analysis showing how performance changes as the relative weighting between physics and data losses is varied? This would provide valuable insight into the method's robustness.\n  - How does PI-DION compare to other methods for solving inverse problems, e.g. Neural Inverse Operators [1]?\n  - Any explanation about why the performance hit between supervised and unsupervised PI-DION is larger for Darcy Flow and Helmholtz equation than for reaction-diffusion?\n\n- How limiting is the assumption that there exists stability estimates for the inverse problem?\n- How well do the theoretical bounds from Theorems 2, 3 match the empirical results of Table 1 (reaction-diffusion and Helmholtz)?\n\n1. Neural Inverse Operators for Solving PDE Inverse Problems", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper engages with an important problem in SciML, learning to solve inverse problems based on physics losses without additional training data.\n- The theoretical results are quite interesting. The authors extend standard stability estimates for inverse problems to the operator learning setting. Promisingly, the theorems apply to the reaction-diffusion equation and the Helmholtz equation, two standard benchmarks in the literature.\n- The proposed method is simple and presented clearly and generally.\n- The empirical results are promising. On three standard benchmarks, the authors demonstrate SOTA performance of supervised learning and near-SOTA of unsupervised learning, compared to supervised DeepONet and FNO.", "weaknesses": "- The main weakness of the paper is that the empirical results, although promising, are relatively limited and could benefit from some clarification:\n  - In Table 1, PI-DION in the supervised learning setting (with 1k training examples) is shown to outperform two different DeepONets and FNOs. However, it's a bit unclear from the paper why this is true, and additional clarification about this would be helpful. Is there a difference in the model architecture / training objective / optimizer between the DeepONets and PI-DION in the supervised setting?\n  - See questions for more.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "CEg1cvslMi", "reviewer": "Reviewer_3Fuq", "summary": "This paper proposes an architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. The architecture  of PI-DIONs is based on DeepONet, and trained with both the physics-infomred loss and data reconstruction loss. The stability estimates established in the inverse problem literature are extended to the operator learning framework. Experiments are conducted to demonstrate the effectiveness of PI-DIONs in learning the solution operators of the inverse problems without the need for labeled data.", "questions": "1. DeepONet and FNO are used for forward problems traditionally, how did they deal with inverse problems in your experiments?\n2. How is the labeled training target f mentioned in line 399 used?  The loss for target f is absent in line 152.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The integration of physics-informed losses into an inverse problem framework based on operator learning is novel, and in principle PI-DIONs can solve the inverse problems (at least in scenarios mentioned in experiments) fast and without the need for labeled data.\n2. Theoretical analysis of the stability estimates is provided.", "weaknesses": "1. Line 243, \"where the term ∥f − f^\\star∥L2(Ωm) in the righthand side\", there is no such term there. Please clarify the equation in line 242 and include all terms on the right-hand side of the equation. \n2. It seems that the input to the reconstruction and inverse branch networks is fixed in shape, corresponding to the partial measurement with given geometry. The observed data in PINNs can have variable count and locations. Please discuss how PI-DIONs might be adapted to handle variable measurement geometries and if there are any limitations on the types of measurement setups it can handle. \n3. In the experiments, PI-DIONs are compared with purely data-driven DeepONet and FNO, which both did not take physics information into account. If possible, please include comparisons with PINNs in the experiments, since both your PI-DIONs and PINNs are physics-informed methods for inverse problems.\n4. The simultaneous training of physics-informed losses for 1000 samples is a difficult task (similar to train 1000 PINNs simultaneously). I am curious about the training difficulties encountered. Please provide specific details on training time, hardware used, and any convergence challenges encountered. If possible, please also include an ablation study on the effect of sample size on PI-DIONs' performance since smaller sample size may lead to easier optimization.\n5. The theoretical analysis on stability estimate is extended from existing key results that considered the single element case. \n6. Please provide a clear definition of u in line 152 and describe its relationship with partial measurement. In line 456, it is better to write \"f(x,y) = 100x(1 − x)y(1 − y) \", so does line 450. \n\nConsidering the above weaknesses, I give a score of 3 to the current version of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "6vYuddPMMR", "reviewer": "Reviewer_knEo", "summary": "This paper introduces Physics-Informed Deep Inverse Operator Networks (PI-DIONs) for solving PDE-based inverse problems without the need for labeled data. The paper extends existing stability estimates from inverse problem literature to the operator learning framework, ensuring the robustness and generalizability of PI-DIONs across the entire function space and domain.", "questions": "See above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper provides a solid theoretical foundation for the proposed PI-DIONs.\n2. The proposed method demonstrates practicality and efficiency in addressing PDE-based inverse problems without the need for labeled data.", "weaknesses": "1. The contribution lacks novelty. The architecture relies on relatively simple components, such as CNNs and MLPs for the branch and trunk networks. It doesn't introduce significant advancements beyond well-established methods.\n2. The baselines used for comparison, such as DeepONet and FNO, are somewhat dated. The paper would benefit from comparisons with more recent and state-of-the-art methods to better demonstrate the model's competitiveness.\n3. The experimental evaluation is limited in range. Conducting experiments on a broader range of benchmarks would strengthen the validation of the proposed method's effectiveness across diverse problems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)