Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data
Authors: Keyan Chen
Yile Li
Da Long
Zhitong Xu
WEI W. XING
Jacob Hochhalter
Shandian Zhe
Venue: Accepted by TMLR ()

Abstract: Neural operators have shown great potential in surrogate modeling. However, training a well-performing neural operator typically requires a substantial amount of data, which can pose a major challenge in complex applications. In such scenarios, detailed physical knowledge can be unavailable or difficult to obtain, and collecting extensive data is often prohibitively expensive. To mitigate this challenge, we propose the Pseudo Physics-Informed Neural Operator (PPI-NO) framework. PPI-NO  constructs a surrogate physics system for the target system using partial differential equations (PDEs) derived from simple, rudimentary physics principles, such as basic differential operators. 
This surrogate system is coupled with a neural operator model, using an alternating update and learning process to iteratively enhance the model's predictive power.
While the physics derived via PPI-NO may not mirror the ground-truth underlying physical laws --- hence the term ``pseudo physics'' ---  this approach significantly improves the accuracy of standard operator learning models in data-scarce scenarios, which is evidenced by extensive evaluations across five benchmark tasks and a fatigue modeling application.

Published in Transactions on Machine Learning Research (11/2025)
Pseudo-Physicc--Inornee Neural Operators: Enhancing Op- erator Learning from Limited Data
Keyan Chen
u1466785@utahh.om edu
Kahlert School of Computing University of Utah
Yile Li*
vile.li@uutah. edu
Kahlert School of Computing University of Utah
Da Long
do. do.long@utah.. edu
Kahlert School of Computing University of Utah
Zhitong Xu
u1502956@uttah.om edu
Kahlert School of Computing University of Utah
Wei W. Xing
w.sing@@sheffilld.com uk
School of Mathematical Physical Sciences University of Sheffield
Jacob Hochhalter
jucob edu
Deportment of Mechanical Engineering University of Utah
Shandian Zhe
zhe@cs. utah. edu
Kahlert School of Computing University of Utah
Reviewed on OpenReview: https: openreview net/ forum? id= 5N1 7D
Abstract
Neural operators have shown great potential in surrogate modeling. However, training a well- performing neural operator typically requires substantial amount of data, which pose major challenge in complex applications. In such scenarios, detailed physical knowledge can be unavailable or difficult to obtain, and collecting extensive data is often prohibitively expensive. To mitigate this challenge, we propose the Pseudo NO) framework. PPI-NO constructs a surrogate physics system for the target system using partial differential equations (PDEs) derived from simple, rudimentary physics principles, such basic differential operators. This surrogate system is coupled with a neural operator model, using an alternating update and learning process to iteratively enhance the model's predictive power. While the physics derived via PPI-NO may not mirror the fround-truth underlying physical laws - hence the term "pscudo physics" - this approach significantly improves the accuracy of standard operator learning models in data-scarce scenarios, which evidenced by extensive evaluations across five benchmark tasks and a fatigue modeling application. Our implementation is released at hhttp:: htt://gittt github. com wwt.c//aaeeeeaanrrrrppeeeeeeeeee
Neural Operator (PPI-

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Published in Transactions on Machine Learning Research (11/2025)
1 Introduction
Operator learning, an important arca for data-driven surrogate modeling, has made significant strides with the emergence of neural operators, which leverage the expressive power of neural networks. Notable examples include Fourier Neural Operators (FNO) Li et al. 2020b), Deep Operator Net (DONet) (Lu et al.) 2021 and other frameworks such as (Cao 2021 Hao et al,) 2023 FNO employs Fourier transform for global convolution and function transformation, while DONet introduces two sub-networks, the branch net and trunk net, to extract representations from the functional space and query locations, respectively, enabling predictions akin to attention mechanisms Vaaswani et al. 2017
For trading for model capacity and performance, neural operators typically require substantial amount of training data to perform optimally. This demand poses challenges, particularly in complex problems, where training data is limited and costly to acquire. In response, the field of physics-infommed machine learning, including physics-informed neural networks (PINN) (Raissi et al.) 2019), has shown promise by incorporating physical laws as soft constraints during training. This approach serves as a regularization technique, embedding fundamental understanding of physics into the model to lessen its reliance on extensive data. Building this idea, the concept of physics-informed neural operators (PINO) has emerged, which integrates physical laws as soft constraints to enhance operator learning while reducing data quantity. It has
been used in Wang et al. 2021 Li et al. 2021 for FNO and DONet training.
Despite the success of PINO, the necessity for a thorough understanding of the underlying physics can pose a significant hurdle, especially in complex applications such as fracture mechanics and climate modeling. Tn those scenarios, the detailed physical knowledge is often unavailable or difficult to identify, and it is often prohibitively expensive to collect extensive data. To navigate these challenges while retaining the benefits of physics-informed learning, we propose the Pseudo Physics-Ifformed Neural Operator (PPI-NO). This framework bypasses the need for exhaustive physical comprehension by constructing partial differential equation (PDE) that characterizes the target system directly from data. The neural PDE is then coupled with the neural operator for alternating updates and training, enabling iterative extraction, refinement and integration of physics knowledge to enhance operator learning. The contribution mainly lies in the following aspects:
workk--newwrk-bbssed
To our knowledge, PPI-NO is the first work to enhance a standard operator learning pipcline using physics directly learned from limited dala, delivering superior accuracy without the need for in-depth physical understanding or extensive data collection.
PPI-NO opens up a new paradigm of physics-informed machine learning where only rudimentary physics assumptions (in this case, the basic differential operations) are required rather than in-depth or rigorous expert knowledge, extending the spectrum of the physics-informed learning for experts of different levels.
The offectiveness of PPI-NO is validated through extensive evaluations on five commonly used benchmark operator learning tasks in literature (Li et al., 2020b Lu et al., 2022), including Darcy flow, nonlinear diffusion, Eikonal, Poisson and advection equations, as well as one application fatigue modeling in fracture mechanics, where the ground-truth holistic PDE system is unknown.
2 Background
Problem Formulation. Operator learning seeks to approximate an operator that maps input parameters and/or functions to corresponding output functions. In many cases, operator learning rises in the context of solving partial differential equations (PDEs), where the operator corresponds to the solution operator of the PDE. Consider a PDE system:
Mx = f(x), x x x (0),
(1)
where x is a compact notation for the spatial and temporal coordinates, Q is the spatial domain, [0, 00) is the temporal domain, A is a nonlinear differential operator, u(x) is the solution function, and fxx is the source term. We aim to learn the solution operator of the PDE system, d F - U where F and U are
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two functional spaces, using training dataset D {ff,; un) which includes different instances of u(-) at set of locations. Once the model is trained, it can be used to directly
and ʃ()) predict the solution function for new instances of the imput f, offering a much more efficient alternative to running numerical solvers from scratch. However, the training datasct still needs to be generated offline using numerical solvers.
Fourier Neural Operator (FNO) (Li al. 2020b) is a popular neural network architecture for operator learning, especially in solving PDEs. For given discretized input function f, FNO first employs a linear layer each component of f at its respective sampling location, thereby lifting the imput into nigher-dimensional channel space. The core of FNO is the Fourier layer, which performs linear transformation followed by a nonlinear activation within the functional space, l(x) + a (Wh(x) + f((( - x (x('))(xx)) where h(x) is the imput to the Fourier layer, s(-) the integration kernel, and (-)) the activation function. The functional convolution computed using the convolution theorem: / [<(x - (x))(xx) = [F(x] F where FF and FF1 denote the Fourier and inverse Fourier transforms, respectively. FNO performs Fast Fourier Transform (FFT) h; multiplies it with the discretized kernel in the frequency domain, and then applies the inverse FFT. After multiple Fourier layers, FNO employs another linear layer to project the latent channels to the original space for prediction.
Deep Operator Network (DONet) (Lu et al., 2021 is another prominent work in operator learning. The architecture of a DONet is structured into two components: a branch net and a trunk net, learning representations for the input functions and querying locations, respectively. Consider an input function f(x) E FF sampled m locations Xg,. xm} and an output function u U. The branch net receives the values [f(x)), (xx),... (xxm)) and outputs feature representation [b1, b.,.,.,, ,bp] TRP. Concurrently, the trunk network processes querying location and outputs another feature vector 12, ,,]p' E R2, The output function value at x is predicted as sum of products of the corresponding elements from the branch and trunk nets, (x) x Ek=1 butR, where G is the learned operator mapping f to u.
Physics-Informed Informed Neural Operator (PINO) (Wang et al. 2021 Li et all 2021 has recently emerged as a promising approach address the data scarcity issue in operator learning. PINO embeds physical laws typically governing equations into the learning process. The incorporation of physics not only enhances the model's adherence to ground-truth phenomena but also reduces its dependency extensive training data. Mathematically, the integration of physics into the learning process can be viewed adding a regularization term in the loss function. Let Ldata represent the standard data-fitting loss term (e.g., the mean squared error between the predicted and actual outputs), and the physics-informed term Lphysics be the residual of the governing PDEs evaluated at the neural network's outputs. The loss function for a PINO model is expressed as
L= Ldata + XCphysies
where À is weighting factor that balances the importance of data-fitting versus physics compliance. This approach encourages the model to learn solutions not only consistent with the provided data but also physically plausible.
3 Methodology
In the absence of physics knowledge (i.e., PDE system equation] D. it is impossible to construct the physics loss term as in PINO. To address this challenge, we propose "pseudo" physics-informed operator learning framework that extracts useful physics representation from data so as to enhance operator lcarning. This framework is motivated by relatively complex applications, where data is often costly and/or limited while the underlying physics hard to fully understand.
3.1 Pseudo Physics System Learning
As the first step, we propose a novel approach to learn the physics system using scarce training data. Our key obscrvation that, although the mapping from to u can be intricate and may necessitate global information across the entire domain (e.g., in linear PDEs, is an integration of the Green's function multiplied with I over the domain), the underlying PDE system equation 1 simplifies to local combination of u and its
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Published in Transactions on Machine Learning Research (11/2025)
14
4
convolution filter
(1)
S,())
S0()) Q+1 Input Channel
sampling location X
S.uu)
sampling location Xj
Q+1 Output Channel
(
S(10(x))
Linear Layer
Linear Layer
Linear Layer
ACT
ACT
f(x))
Figure 1:
The illustration of "pseudo" physics representation network The input consists of and its finite Sq (u){) across different sampling locations. The top row shows a
difference derivative approximations (u). convolution layer that aggregates local neighborhood to compensate the information loss caused by finite difference. The bottom row shows that & uses fully connected layers at each sampling location to combine and its derivatives locally predict f at the location.
derivatives. We therefore design neural network to approximate the general form of N,
Mu]((x x o (x,
Sq()((x)))
(2)
where { are Q derivative operators that we believe should be present in the system, such as ue d Üz, u, Ozg u, 02,2, u, 0z u, Ozgzz u, and more.
The inherent local combinatorial nature of the PDE representation decouples the values of and its derivatives across different sampling locations, thereby significantly reducing the learning complexity and the amount of training data required. For instance, consider sampling the input function and output function on 128 x 128 grid. A single pair of discretized input and output functions, denoted as u), is typically insufficient for training a neural operator, because learning the mapping f - u requires abundant global information. However, this pair can be decomposed into 128 x 128 = 16, 384 training data points across various (spatial and temporal) locations to train & outlined in equation 3 We use cach point of the grid to construct training sample. This decomposition provides rich information about the local relationships between those derivatives. Therefore, learning of the PDE system / through our formulation in equation can still yield promising accuracy in predicting f()) as in equation
with small number of (f,, u) pairs, hypothesize that the
We use an Ly loss to estimate the paramoters ø, which is defined as
N
M
o(x,, un (x)), S1 (xx) SQ (x())((x))
[x(xx))
(3)
where fa(+) and (.). are the input and output functions in n-th training example, and (xi,... [... are the locations at which we sample fn and un.
We use finite difference to obtain the derivatives of each un namely, S*(u,) (1 < Q), and then pass these inputs to the network to compute the prediction. Specifically, employ centered finite differences to
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Φ
NO
2
I
Figure 2: PPI-NO learning framework.
approximate the derivatives of a classical scheme with well-established accuracy and sample complexity. The approximation error scales as O(h²) for all derivative orders, where h is the grid spacing. To achieve an error < 6, the required number of grid points is 0(c=d/2), where the input dimension (Mitchell & Gritfiths p980h. It worth- nottin that while we can separate out training examples across grid points, these examples are not independent to each other. The effective sample size (ESS) for training our ø-network is smaller than the total number of grid points. To compensate numerical errors of the finite difference and leverage deterddependencies between the training examples, we incorporate convolution layer to integrate the neighborhood information (see Fig. top). Let (u,) represents the finite difference approximation of S& (u,) (1 & l < Q): for k = O, we define Sx(un) = un- Each (u(u)) is treated as an input channel. After applying the convolution, the output at each sampling location xj for channel is given by
(x(x)==
Σ
w. (x(Ar))S(xx)).
(4)
c= x. Enci(xy)
where nei(-) denotes the neighborhood sampling locations defined in the convolution filter, Ai is the relative distance between x; and Xj, and w. (Air) is the corresponding filter weight. Each Sx(x,) can be viewed as an interpolation, providing a new approximation of S&(x,) by incorporating all neighborhood values of un (xj) and their finite difference derivative approximations. The convolution results are then passed into subsequent layers. The interpolation (filter) weights are jointly learned from data. This design allows us integrate additional information into the inputs facilitate the learning of ø. Empirical results confirm that incorporating this convolution layer improves the prediction accuracy; refer to the ablation study in Section 5.2 and Table 7a Next, at each sampling location Xj, we employ fully connected layers (i.e., ML.P) to combine all {Sx(x,)]E=o in an arbitrarily flexible way to predict f(x;); see Fig. bottom.
As neural network, we note that the o-network, has sufficient representation power approximate any PDE operator of the form
L(x, ((u)))), SQ(w((z)),
where S; denotes derivatives, e.g., L Urx + uu, + n² + cos(u). Because most practical PDFs are smooth analytic functions of u and its derivatives, the universal approximation theorem guarantees that such mappings can be accuratoly approximated by a food-forward network of the following form:
f(x) Σ oi owT x bb),
given sufficient neurons and non-polynomial activation (Cybenko 1989 Hornik 1991
The sample
complexity m(c,5) is the smallest number of samples m ensuring, with probability at least - 8,
R(h) R(h)) s, Yh E H.
where R(h) and R(h) denote generalization and training errors, respectively. From computational learning theory (Shalev-Shwartz & Bavid 2014
m(c, = (dvc
(dvc + log(1/6)
2
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Published in Transactions on Machine Learning Research (11/2025)
where dyc is the VC dimension of the network. For network with W parameters and piccewise-linear activations (e.g., ReL.)), dyc O( log W); for smooth activations, dvc O(W2). Therefore, the learned neural network mapping ø + f, although black-box in nature, is expressive enough to encapsulate valuable physics knowledge inherent in the data employed for operator learning.
Our method can be readily adapted other numerical approaches for derivative approximation. For instance, when functions are irregularly sampled, smooth function estimators such kernel interpolation (Long et al, 2024 the RBF-FD method (Fornberg et al., 2013 Fornberg & F Flyer 2015 Bayesian B-splines (Sun et al, 2022 can be used to estimate gradient information directly from data. These estimated gradients can then serve as inputs to our PDE neural network & for further training.
3.2 Coupling Neural Operator with Pseudo Physics
Next, we leverage the pseudo physics laws embedded in the learned mapping o: - to enhance neural operator learning. Specifically, we use e to reconstruct. f from the u predicted by the neural operator. In this way, our approach uses the physics learned in the previous step to incorporate reconstruction error into the learning of the neural operator parameters; see Fig. 2 for an illustration.
Initially, we train the neural operator 4b + u using the available training data, creating a preliminary model. This model is developed using FNO or DONet or other neural operators. The focus is to first establish basic understanding of the relationship between f and from the limitod data. Next, the loss function for d is augmented using the physics laws learned in the first step,
N
L Σ n=1
(o(V(f)), + + Epipy --Bpr
where the first term is the C2 loss for data fitting (as in the standard neural operator training), and the second term is the expected reconstruction error for the input function. The second term incorporates the physics laws embedded in o(-), and X is a weighting factor that balances the training data loss against the reconstruction error.
In practice, the expected reconstruction error does not have a closed form. One can sample a collection of f' from the underlying distribution of the input function p(.), e.g., a Gauss random field or Gaussian process, and then employ a Monte-Carlo approximation,
1 N E o(U()))
C= Σ (2(b(())
(5)
where N' is the number of input function samples. To be more specific, the PDE network á acts purely as a model-based regularizer, without introducing any additional labeled data. In practice, it realistic to sample as many as possible new input f' from the same input distribution, but it often not realistic acquire all the corresponding ground-truth solutions u'. Thus, we sampled f', pass through the current operator network v obtain predicted solution á', and then feed ú' into the PDE network to produce reconstructed source term f'. We use the discrepancy I| f'122 as regularization term, as formalized in equation 5 It simply encourages w's predictions to lie on the manifold of solutions that are consistent with the PDE representation vb
To enhance the operator learning process, the model is iteratively refined. each iteration, we first fine-tue the neural operator & with the pscudo physics & fixed, and then fix v, finc-tune to refine the physics representation. This fine-tuning loop is carried out for multiple iterations, allowing for continuous improvement of the neural operator based the refined physics representation. Our method is summarized in Algorithm 1

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Published in Transactions on Machine Learning Research (11/2025)
Algorithm 1 Paeudlo-Ptyysics-lnforr NO
1: Train a preliminary NO & with standard NO loss.
2: Train a preliminary psuedo physics network o with loss Ls in equation
3: repeat
4: Sample N' instances from the input function space.
5: Fix pseudo physics network o, fine tune NO & with the loss equation 5
Fix NO v, fine tune the pseudo physics network & with the loss equation
until Maximum iterations are donc or convergence
4 Related Work
Operator learning is a rapidly evolving research field, with a variety of methods categorized as neural operators.
Alongside FNO, several notable approaches have been introduced, such as low-rank neural operator (LNO) (Li
ie al.
2020b,, multipole graph neural operator (MGNO) (Li et al. 2020a). multiwavelet- based NO (Gptta
ct al.
2. D, and convolutional NOs (CNO) (Raonic et al. 2023)- Deep Operator Net (DON) (Lu al.
2021 is another popular approach, consisting of a branch network applied to input function values and a
trunk network applied to sampling locations. The final prediction is obtained through the dot product of
the outputs from the two networks. To enhance stability and efficiency, Lu et al. (2022) proposed replacing
the trunk network with POD (PCA) bases. Recently, transformer architoctures have also been employed to
design neural operators, E.9,,
(Cao 2021
Li el al. 2022: Hao el al. 2023).
Physics-Informed Neural Networks (PINNs) RRassi et al. 2019 mark a significant advancement in scientific machine lcarning. PINNS integrate physical laws directly into the learning process, making them effective for solving differential equalions and understanding complex physical systems. This methodology is particularly beneficial in scenarios where data is limited or expensive to obtain. Li et al. (002 introduced a dual-resolution approach that combines low-resolution empirical data with high-resolution PDE constraints. This method achieves precise emulation of solution operators across various PDE classes. In parallel, physics-informed DONet by Wang et al. 2021 incorporate regularization strategies enforcing physical law adherence into the training of DONets. Zanardi et al. (0233) presented an approach using PINO for simulations in non- informed
equilibrium reacting flows. Lee al. (2023) proposed opPINN, a framework combining physics-informed Landau (FPL) equation. [Rosofsky et al. 2023) provided a review of applications of physics-informed neural operators. However, existing methods demand one must know the physics laws beforehand, which might not be feasible in many practical applications or complex systems. Our method offers a simple and effective framework, enabling the extraction of implicit physics laws directly from data, even when the data is scarce. Empirically, these pseudo physics laws have proven to be highly beneficial in enhancing the performance of operator learning, as demonstratod in Section 5 Many methods have been developed specifically for discovering differential equations from data, including SINDy (Brunton et al., (0166) and its variants (Schaeffer) [0117] Zhang & Ma 2020 Lagergren et al., 2020). PINN-SR (Chen et al.] 2021), Bayesian spline learning (Sun et al. 2022). and kernel-based equation discovery (Lngg et al. 2024). To ensure interpretability, these approaches typically assumee specific equation form and perform sparse selection from a set of candidate operators. In contrast to these methods, which prioritize interpretability, our approach focuses on enhancing the prediction accuracy of operator learning under limited data. To this end, we utilize a black- box neural network to represent PDEs. While this offers greater flexibility, it comes at the cost of reduced interpretability. Our method employs an alternating update strategy to jointly refine the PDE representation and improve operator learning.
neural networks with operator learning for solving the Fokker-Planck- Llane---a
Our work is also related to the cycle consistence framework (Zhu et al. 2017 for image-to-image translation. A critical difference is that cycle-consietence performs unpaired image- taaage-to-image translation, while our method aims for accurate paired translation (mapping). In cycle-consistence, the translation is viewed successfully as long as the translated images follow the target distribution. Hence, cycle-consistence has a much more relaxed objective. Another key difference is that our method aims to improve the learning of a function-to- function mapping with very limited data- that motivates us learn a "pscudo" " physics representation. The cycle-consistence framework relies on adversarial training which typically requires a large amount of data to obtain successful learning outcomes.
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2b
t
(x, y)
Y
X
2c
Figure 3: Example of semi-elliptic surface crack on a plates MMrrell et al. 2024
5
Experiments
We tested five commonly used benchmark operator learning problems in the literature Li et al. 2020b Lu et al. 2022), including Darcy Flow, Nonlincar Diffussin,, Eikonal, Poisson and Advection. In addition, we examined our method in an application of fatigue modeling. The task is to predict the stress intensity factor (SIF) for semi-elliptical surface cracks on plates, given three geometric parameters that characterize the cracks (Merell et al. 2024); see Fig. 3 The SIF plays critical role in modcling crack growth by quantifying the stress state near the tip of a crack, and hence SIF computation and analysis are extremely important in fatigue modeling and fracture mechanics (Anderson & Anderson 2005). The STF computation is expensive, because it typically needs to run finite element method (FEM) extended FEM with very fine meshes (Kuna 2013 Due to the complex sequence of computational steps involved in SIF calculation, there is no holistic PDE that directly models the relstionship between the geometric features and the SIF function. Instead, SIF computation typically relies on numerical methods and the extraction of local stress fields near the crack tip. The details about all the datasets are given Section A of the Appendix.
We evaluated our method based on two widely used NO models, FNO and DONet. Note that our method is straightforward to implement on other NO models, such attention based models.
For each operator learning benchmark, we simulated 100 examples for testing, and varied the number of training examples from {5, 10, 20, 30), except for Advection, we ran with {20, 30, 50, 80} training examples. For SIF prediction, which is much more challenging, we experimented with training size from (400, 500, 600), and employed 200 lest examples. We repeated the evaluation for five times, each time we randomly sampled a different training set. The input to the pseudo physics networks includes all the partial derivatives up to the 2nd order. For the pseudo physics neural network d see equation 3 we tuned the kernel size from {33, 3), (5, 5), (7, 7), (9, 9)}. The stride set to 1 and padding shape does not change. In the subsequent fully connected layers, we chose the number of layers from (3, 4, 5, 6), and the layer width from {16, 32, 643. We used GeLU activation. For FNO, we set the number of modes to 12 and channels to 32 (in the lifted space). We varied the nunber of Fourier layers from {2, 3, 4). For DONet, in all the cases except Darcy Flow, the trunk net and branch net were constructed as fully connected layers. We varied the number of layers from {2, 3, 4} and the layer width was chosen from (30, 40, 50, 60), with ReLU activation. For the case of Darcy flow, we found that DONet with only fully connected layers exhibited inferior performance. To address this, introduced convolution layers into the branch net. We selected the number of convolution layers from {3,5,7}, and employed batch normalization and leaky ReLU after each convolution layer. To incorporate the learned pscudo physics representation into the training of FNO or DONet, we randomly sampled 200 input functions lo construct the second loss term in equation 5 We set the maximum number of iterations to 10 and selected the weight À from [10-1, 110]]. All the models were implemented by PyTorch (Paszke et al. 2019),, and optimized with ADAM (Kingma Ba 2014). The learning rate was selected from {10- 5 10 10- 3). The number of epochs for training or fine-tuning FNO, DONet and pseudo physics network was set to 500 to ensure convergence.
set to "same" to ensure the output
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Table 1: Relative L2 error five operator learning benchmarks, where "PP[" short for "Pseudo-Physiie Informed". The results were averaged from five runs.
(a) Darcy flow
Treining size
5
20
30
FNO
0.4915 + 0.0210
0.3870 + 0.0118
0.2783 = 0.0212
0.1645 + 0.0071
PPI-FNO
0.1716 + 0.0048
0.0956
0.0084
0.0680 = 0.0031
0.0642 = 0.0010
Error Reduction
65.08%
75.29%
75.56%
60.97%
DONet
0.8678 + 0.0089
0.6854 + 0.0863
0.5841 = 0.0279
0.5672 = 0.0172
PPI-DONet 0.5214 + 0.0543
0.3408
0.0209
0.2775
0.0224
0.2611 = 0.0084
Error Reduction
39.91%
50.27%
52.49%
53.96%
(b) Nonlinear diffusion
Training size
FNO
0,2004 + 0.0083
0.1242 + 0.0046
0.0876 +: 0.0061
0.0551
0.0021
PPI-FNO 0.0105 + 0.0016
0.0066 + 0.00023
0.0049
0.00037
0.0038 + 0.00039
Error Reduction
94.76%
94.68%
94.40%
90%
DONet
0.3010 + 0.0119
0,2505 = 0.0057
0.1726 = 0.0076
0. 041430 + 0.0036
PPI-DONet 0.1478# 0.0126
0.1161 + 0.0124
0.1032
0.0059
0.0842 0.0041
Error Reduction
50.89%
53.65%
40.20 %
41.11%
(c) Eikonal
Training size
5
10
20
30
FNO PPI-FNO 0.0678 + 0.0026
0.2102 + 0.0133
0.1562 0.0098
0.0981 + 0.0022
0.0843 = 0.0020
0.0582 + 0.0043
0.0493 + 0.0023
0.0459 + 0.0010
Error Reduction
67.74%
62.74%
49.74%
45.55%
DONet
0.3374 + 0.0944
0.1759 + 0.0065
0.1191 1 0.0047
0.1096 + 0.0037
PPI-DONet 0.1302+ 0.0127
0.0907 + 0.0093
0.0714 + 0.0011
0.0700 + 0.0007
Error Reduction
61.41%
48.43%
40.05%
36.13%
(d) Poisson
Trwining size FNO
5
10
20
30
(.22440 + 0.0083
0.1390
0.0007
0.0895
0.0008
0.0698 + 0.0014
PPI-FNO 0.1437 + 0.0062
0.0771 + 0.0018
0.0544 = 0.0009
0.0458 + 0.0003
Error Reduction
38.59%
44.53%
39.22%
34.38%
DONet
0.6142 = 0.0046
0.5839
0.0090
0.5320 = 0.0028
0.5195 = 0.0040
PPI-DONet
0.5275 + 0.0037
0.5001 + 0.0042
0.4450 = 0.0010
0. 0.4258 = 0.0040
Error Reduction
14.12%
14.35%
16.35%
18.04%
(e) Advection
Truining size FNO PPI-ENO Error Reduction
20
30
50
80
0.4872 + 0.0097
0.4035
0.0086
0.3019
0.0085
0.2482 1 0.0059
0.3693 1 0.0099
0.3224 1 0.0123
0.2236 1 0.0075
0.1698 1 0.0075
24.20%
20.10%
25.94%
31.59%
DONet
0.5795 + 0.0045
0.4810 + 0.0092
0.3882 = 0.0086
0.3164 - 0.0072
PPI-DONet
0.3630 + 0.0112
0.2897 + 0.0097
0.2629 = 0.0053
0.2120 + 0,0065
Error Reduction
37.36%
39.77%
32.28%
33.00%
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5.1 Predictive Performance
Table 2: SIF prediction error for plate surface cracks in fatigue modeling.
Training size
400
500
600
FNO
0.1776 = 0.0150
0.1695 + 0.0090
0.1122 + 0,0094
PPI-FNO 0.1166
0.0064
0.1151 + 0.0093
0.0850=000000
Error Reduction
34.35%
32.09%
24.24%
DONet
0.5318 = 0.0095
0.5155 + 0.0200
0.4037 + 0.0331
PPI-DONet
0.3490 + 0.0034
0.3468 + 0.0074
0.3299 + 0.0066
Error Reduction
34.37%
32. 73%
18.28%
Table Relative L2 error of using the learned back-box PDE network equation
predict the input function f.
(a) Training size-10
Benchmark
MLP
Ours
Darcy Flow
D. 0.1819#0.0028
0.1392+ 0.0080
Nonlinear Diffusion
0.0660:10.006 0.0144:10.0099
0.0233
Eikonal
0.0108 0.0006
(b) Training size=30
Benchmark
MLP
Ours
Darcy Flow
0.001330.0013
0.06881 0.0032
Nonlinear Diffusion
0.0463+000022
0.0163+0.0002
Fikonal
0.0070=0.00005
0.0052 + 0.0002
We reported the average relative Le error and the standard deviation (with and without using pseudo physics informed learning) in Table I and Table The model trained with the pseudo physics network (see equation 5 is denoted as PPI-FNO or PPI-DONet, short for Pseudo Physics Informed FNO/DONet. As shown, across all the cases, with our pseudo physics informed approach, the prediction error of both FNO and DONet undergoes a large reduction. For instance, across all training sizes in Darcy Flow and nonlineur diffusion, PPI-FNO reduces the relative L2 error of the ordinary FNO by over 60% and 93%, respectively. In Darcy Flow with training sizes 10 to 30, PPI-DONet reduces the error of the ordinary DONet by over 50%. In SIF prediction, our method applied to both FNO and DONet reduced the error by over 30% for training sizes of 400 and 500. Even the minimum reduction across all cases achieves 14.12% (PPI-DONet over DONet on Poisson with training size 5).
Together these results demonstrate the strong positive impact of the learned physics by our neural network model e specified in Section 3.1 Although it remains opaque and non-interpretable, it encapsulates valuable knowledge that substantially enhances the performance of operator learning with limited data.
Next, assessed the accuracy of the learned physics laws by examining the relative L2 error in predicting the source functions ff from è (see equation 3). We tested Darcy Flow, nonlinear diffusion, and Eikonal. We compared a baseline method that removes the convolution layer of é leaving only the fully connected layers, namely MLP (see Fig bottom). The results are reported in Table 3 It can be observed that in nearly every case, adding a convolution layer indeed significantly improves the accuracy of Φ. This improvement might be attributed the convolution layer's ability to integrate neighboring information and compensate for the information loss introduced by finite difference approximating the derivatives. We also experimented with multiple convolution layers, but the improvement was found to be marginal.
In addition, found the operator learning improvement is relatively robust to the accuracy of our physics representation é. For instance, on Darcy Flow with training size 5 and 10, the relative L2 error of é network is 0.2285 and 0.1392, which is significantly bigger than with training size 30 where the relative L2 error is 0.0688. Yet the error reduction upon FNO (see Table Ia] under all the three training sizes above 60%. The error reduction upon DONet is 40% for training size and over 50% for training size 10 and 30. The results imply that even roughly capturing the underlying physics (with Φ) can substantially boost the operator learning performance.
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Table 4: Relative Le error in five operator learning benchmarks with richer data, where "PPI" is short for "Pseudo- Physics Informed". The results were averaged from five runs.
(a) Darcy flow
Training size
600
1000
FNO
0.0093 + 0.00015
0.0079 + 0.00018
PPI-FNO
0.0087 + 0.00040
0.0082 + 0.00039
DONet
0.0540 = 0.00064
0.0446 + 0.00023
PPI-DONet
0.0415 + 0.00077
0.0362 +: 0.00049
(b) Nonlinear diffuusion
Training size
600
1000
FNO
0.0035 + 0.0007
0.0028 = 0.000-1
PPI-ENO
0.0047 1 0.00102
0.0042 1: 0.00096
DONet
0.0222 + 0.00020
0.0187 1 0.00023
PPI-DONet
0.037911 0.00088
0.0368 + 0.00093
(c) Eikonal
Training size
600
1000
FNO
0.0193 + 0.00017
0.0148 + 0.00009
PPI-FNO
0.0199 + 0.00016 6
0.0160 + 0.00009
DONet
0.0460 + 0.00021
0.0411
0.00038
PPI-DONet
0.0175-t 0.00047
0.0436 + 0.00031
(d) Poisson
Training size
600
1000
FNO
0.0045 + 0.00006
0.0037 + 0.00005
PPI-ENO
0.00416 + 0.00005
0.0040 + 0.00006
DONet
0.1786 +. 0.00398
0.1719
0.006443
PPI-DONet
0.1409 + 0.00292
0.1373 1 0.00136
(e) Advection
Training size
600
1000
FNO
0.0943 + 0.00177
0.0768 +: 0.00182
PPI-FNO
0.0819 + 0.00092
0.0695 + 0.00092
DONet
0.0913 + 0.00074
0.0748 1 0.00098
PPI-DONet
0.0732 + 0.00143
0.0626 + 0.00107
Table 5: Relative La error in Poisson and Advection operator learning benchmarks, where "PPI" is short for "Pseudo-Physics Informed" and "PI" is truly "Physics Informed". The results were averaged from five runs.
(a) Poisson
Training size
10
20
30
FNO
0.2340 + 0.0083
0.1390 = 0.0007
0.0895 + 0.0008
0.0698 = 0.0014
PPI-ENO
0.1437
0.0062
0.0771 = 0.0018
0.0544
0.0009
0.0458 = 0.0003
PLENO
0.1433 +. 0.0104
0.0718 1 0.0015
0.0504 1 0.0009
0.0429 + 0.0004
(b) Advection
Trasning size FNO
30
80
0.4872
0.0097
0.4035 + 0.0086
0.3019
0.0085
0.2482 +: 0.0059
PPI-ENO
0.3693
0.0099
0.3224 +: 0.0123
0.2236
0.0075
0.1698 1: 0.0075
PI-NOO
0.3628 0.0082
0.3205 = 0.0121
0.2222 0.0074
0.1668
0.0057
For a further assessment, we conducted a fine-grained evaluation by visualizing the predictions and point-wis wise errors made by each method. In Fig. 4a and [bb] we showcased the predictions and point-wise errors using PPI-DONet for Dorey Flow, PPI-FNO for nonlineor diffusion, respectively. Additional examples of predictions
and point-wise errors are provided in Fig. 5a
Ga
6b and
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eNOoerrar
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PPL-ENO-pred
eNOoecror
PPI-FNO etror
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0,00 0,0015
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DONet-pred 0.90)
PEL-DONd-prod
DONet-erorr
POL-DONet-ator
Ground-truhh
FNO-pred
PPI-FNO-prod
ENO-eror
PPL-FNO-stror
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Ground-truh
PPL-DONd-prred
Ground-truth
FNO-pned
PPI-ENO-peod
FNO-emor
PPL-ENO-aRor
(a) PPI-DONel: Darcy Flow
(b) PPI-ENO: nonlinear diffusion
Figure 4: Examples of the prediction and point- wise error PPI-DONet and PPI-FNO on Darcy Flow and nonlinear diffusion, respectively. From top to bottom, the models were trained with 5, 10, 20, 30 examples.
It is evident that without the assistance of the pseudo physics laws learned by our method, the ordinary DONet and FNO frequently missed crucial local structures, sometimes even learning entirely incorrect structures. For example, In Fig. 4a the first row, DONet missed one mode, while in the second and third row of Fig. 4a DONet failed to capture all the local modes. After incorporating the learned physics, DONet (now denoted as PPI-DONet; see the third column) successfully captures all the local modes, including their shapes and positions. Although
all the details are exactly recovered, the point-wise error is substantially reduced, particularly in those high error regions of the ordinary DONet; see the fourth column of Fig. 4al In another instance, shown in Fig. 4b where the ordinary FNO (second column) captured the global shape of the solution, but the missspecification of many local details led to large point-wise errors across many regions (fourth column). In contrast, PPI-FNO (third column) not only identified the structures within the solution but also successfully recovered the details. As a result, the point-wiss wise error (fifth column) was close to zero everywhere. Additional instances can be found in Fig. [a the first three rows illustrate that ordinary FNO (trained with 5, 10, and 20 examples, respectively) estimates an entirely incorrect structure the solution, indicating that the training data insufficient for FNO capture even the basic structure of the solution. In contrast, after joint training with physics network from the out the solution structures and yielded substantial reduction in point-wise error across nearly everywhere. The point-wise error became uniformly close to zero. With 30 examples, the ordinary FNO was then able to capture the global structure of the solution, but the details in the bottom left, bottom right, and top right corners were incorrectly predicted. In comparison, PPI-ENO further recovered these details accurately.
sparse data, PPI-FNO accurately figured
Collectively, these results demonstrate that the pseudo physics extracted by our method only substantially boosts the overall prediction accuracy but also better recovers the local structures and details of the solution.
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50
100
150
50
100
301
100
Ground- mith
FNO-pred
PPL-FNO-pred
FNO-emor
PPI-FNO-enor
Ground-truth
DONet-pred
PFI-DONd-pred
DONd-arror
100
20
40
Ground-eruh euth
prod
PPI-PNO-pred
PNO
PPI-FNO error
Ground-tuuh
DONet-pred
PEI-DONGl-pred
DONet-ctoo
PFI-DONd-stror
Ground--ruth
FNO-pred
PPI-FNO-pred
ENO-erior
PPI-FNO-aror
Ground-trut truth
DONet-pred
PPI-DONet-pred
DONet-oroo
PPI-DONet- Net--rror
100
Ground-truth
FNO-pred
PPI-PNO-gred
FNO-crror
PPI-FNO-error
Ground-truth
DONe--pred
PPL-DONeieped
D'NNe--orr PFI-DONdd-aror
(a) PPI-PNO: Eikonal
(b)
PPI-DONet: Eikonal
Figure 5: Examples of the prediction and point- wise of PPI-FNO and PPI-DONet Bikonal. From top to bottom, the models were trained with 5, 10, 20, 30 examples.
5.2 Ablation Studies
We additionally evaluate stepwise pipeline in which a standard system-identiiiiatioon method is first used to infer an explicit PDE model, and the resulting equation is then used to regularize neural-operator training. However, system identification methods targeting interpretability typically require stronger assumptions and more prior knowledge than our black-box PDE Icarner. For instance, the widely used SINDy framework (Brunton et al., 2016) assumes a linear structure and depends on a predefined operalor dictionary whose choice crucial and can easily cause model misspecification.
Step-by-step approach:
In contrast, our black-box PDE representation is more flexible and assumption-lihht: only specify set of basic derivatives, while their nonlinear combinations (e.g., sin(u), uun,, u² Uy are learned by neural network. This design inevitably sacrifices interpretability in exchange for flexibility. Furthermore, our joint training of the neural operator and the PDE representation allows both components to mutually enhance each other's performance.
evaluate the "syston-identification htt.riiiiiiiiiio----oooooaaaaaaa noural-operator" altornative, we ran SINDy first identify an approximale governing equation, used to generate paired synthetic data, and then trained the FNO on the combined real and synthetic samples. The operator dictionary matched the derivative set used in our method. For comparison, we also ran our black-box ø-network to generate 200 synthetic examples. The resulting test errors of the neural operator under varying training data sizes are reported in Table 6
We observe that applying SINDy before training FNO improves prediction accuracy only in the Darcy flow task with 5, and 20 training examples; in all other cases, performance actually deteriorates compared training FNO alone. This suggests that the recovered equations, while interpretable, may still be inaccurate enough that the synthetic samples they generate adversely affect FNO training.
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0.000
0,01
0,00005 0,00050 0.00075
0.00
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FND-goed
PPI-FNO-pred
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PPI-FNO-eron
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PPI-DONet-pred
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PPI-DONet-eroo
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PPI-EXO-pred
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PPI-ENO-eTve
Ground truth
DONet-pred PPl-DONee-pred DONet-error
PPt-DONee-erro
(a) PPI-FNO: Darcy Flow
(b) PPI-DONet: Nonlinear Diffusion
Figure 6: Examples of the prediction and point-wise error of PPI-ENO and PPI-DONet Darey Flow and Nonlinear diffusion, respectively. From top to bottom, the models were trained with 5, 10, 20, examples.
35.0
Ground Truth
35.0
Grvund Truih
35.0
PÑO
Ground Truth
32.5
32.5
32.5
27.5
27.5
25.0
22.5
20.0
20.8
Figure 7: Examples of SIF prediction of FNO and PPI-FNO trained with 600 examples.
In contrast, the "d-networkk-hen- pipeline"- where our d-network first generates synthetic data, followed by FNO training consistently improves results. Although the d-network is a black-box model, it can more accurately approximate the underlying PDE structure, leading to more useful synthetic samples. Nevertheless, this sequential approach still underperforms our "jointly trained & + FNO" model, demonstrating that mutual optimization between the two components further enhances FNO's predictive accuracy.
Training with rich data. Although our method is designed to enhance operator learning with limited data, also evaluated its performance when the training data is abundant. Specifically, we increased the training size of the five operator learning benchmarks to 600 and 1000 examples. The results, presented in Table show that the accuracy of the standard NO and our method, PPI-NO, becomes comparable. For Darey flow and Advection, our method achieves a slight improvement, whereas for Eikonal and Nonlinear diffusion, it performs slightly worse. This may be bocause, with sufficiently rich data, the standard NO can alrcady utilize the data effectively to achieve good performance, leaving little room for the added value of the learned physics to further improve the results. In such cases, the benefit of incorporating physics knowledge becomes
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Table 6: The relative Lg error with using syterma-deeiiiiccttoo---eeerrrpp on Darcy flow and Nonlinear Diffusion benchmarks.
(a) Predicting u via different architectures Darcy flow.
Training size
5
10
20
30
SINDy + FNO
0.4624=0.0147
0.3382=0.0013
0.2725=0.0115
0.2222=0.0033
networkk - FNO
0.4192=000196
0.2638=0.0132
0.1616=000018
0.1013=:0.0023
FNO
0.1115=000210
0.3870=0.0118
0.2783=0.0212
0.1615==0.001
Ours
0.1716=0.0048
0.0956=0.0084
0.0680=0.0031
0.0642=0.0010
(b) Predicting u via different architectures on Nonlinear Diffusion.
Training size
20
30
SINDy FNO
0.2703=0.0045
1.2400=000005
0.1908=0.0123
0.1213=0.0034
ø-network- - FNO FNO
0.1603=0.0108
0.0943#0.0083
0.0540=0.0043
0.0361#0.0013
0.2004110.0083 0.1242110.0046
0.087610.0061
0.055110.0021
Ours
0.010511.0016
0.0066100.0023
0.004910.00033
0.003810.00039
marginal. Furthermore, our alternating updating mechanism brings additional optimization workload, which may introduce additional complexity to the learning process.
Comparison with NO training using ground-truth physics. Our work is motivated by scenarios where the underlying physics is unknown. It is therefore insight,uu to compare our approach with training NO using the ground-truth physics, as done in PINO. This comparison allows us to examine how our learned "pseudo" physics differs from the true physics in facilitating operator learning. To this end, we tested PINO on the Poisson and Advection cases, using finite difference approximations to compute the derivatives. We used FNO the baseline. The PDE residual was incorporated into the training loss. As shown in Table 5 while PINO consistently achieves even better results, the performance of our method is close to PINO's. This demonstrates that our learned black-box representation, despite lacking interpretability, can provide a similar benefit in improving operator learning performance.
Learning behavior. We examined the learning behavior of our method, which conducts an iterative, alternatingly fine-tuning process. We employed one Darcy Flow, one nonlinear diffussion and one Eikonal dataset, each with 30 examples. We show the test relative L2 error along with the iterations in Fig. 8 As we can see, the predictive performance of our algorithm kept improving and tended to converge at last, affirming the efficacy of our learning process.
Ablation study on the weight A. We examined the effect of the weight À of our "pseudo physics"; see equation 5. To this end, we employed Darey Flow, nonlinear diffusion, and Eikonal, each with 30 examples for training. We varied À from [0.5, 10a), and run PPI-FNO and PPI-DONet on these datasets. From Fig, g we can see that across a wide range of values, PPI-FNO and PPI-DONet can consistently putperform the standard FNO and DONet respectively by a large margin. However, the choice of does have a significant inffuence on the operator learning performance, and the best choice is often in between.
Ablation study on pseudo physics network d. To confirm the efficacy of our designed network & in facilitating operator learning, we considered alternative designs for di (1) using standard FNO to predict f directly from u; no derivative information is included in the input; (2) removing the convolution layer in our model, and just keeping the fully-connected layers, namely MLP; the derivative information not included in the input. With different designs of Φ, we evaluated the PPI learning performance on the Darcy Flow benchmark. The relative La errors in predicting f via and predieting u are reported in Table 7 It can be seen that our design of & consistently outperforms alternative architectures by a notable margin, showing the effectiveness of learning a (black-box) PDE representation and improving the operator learning.
It worth noting that the PDE network & is trained on exactly the same input distribution as the main operator network and all the baselines. We neither extrapolate beyond the original spatial domain nor alter the distribution over coefficients or initial conditions. When we refer to "randomly sampled inputs", these are drawn from the same domain and input distribution as the training data and are used solely as additional collocation points for evaluating the pseudo-PDE residual, rather than as new labeled pairs. In particular,
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since only imput functions f' are sampled, they cannot be used as (f, u) pairs in the original neural-operator training.
Table 7 also includes a variant in which we simply employ an FNO as a drop-in replacement for the d- network, trained under the same input distribution. This variant is consistently much worse than our design, indicating that the performance gains do not stem from providing the PDF network with a larger or richer input distribution, nor from increased model capacity alone, but rather from the specific PDE-structured regularization induced by our é-network.
Table 7:
The relative L2 error with using different architectures of & in pendo--physsee--nnforeed (PPI) learning Darey Row benchmark.
(a) Predicting via ø with different architectures.
Training size
10
20
30
FNO
0.7229=0.0315
0.5759+ 0.0126
0.4257#
0.0106
0.3160:: 0.0037
MLP
0.7169-10.0100
0.659811 0.0056
0.64641
0.0029
0.627711
0.0032
Ours
0.2285 1 0.0147
0.1392 1 0.0080
0.0898
0.0046
0.0688 1 0.0032
(b) Predicting u.
Training size PPI-ENO with FNO as & PPI-FNO with MLP as Φ
20
30
0.5853.10.0155
0.38711 0.0124
0.261331 0.0190
0.16291 0.0064
0.7262=40.0920
0.5516# 0.0699
0.4568+ 0.0857
0.3983+ 0.1051
Standard FNO Ours
0.4915 + 0.0210
0.3870 + 0.0118
0.2783 + 0.0212
0.1645 + 0.0071
0.1716 + 0.0048
0.0956 + 0.0084
0.0680 + 0.0031
0.0642 + 0.0010
Table The relative La error of PPI learning by incorporating different orders of derivatives. During the comparison with other operator learning methods, used derivative orders up to 2 to run our method.
(a) Predicting f via Φb
Trasning size
5
order 0
0.7126=0.0131
0.5733=:00000
0.4812::0.0399
0.3445-10.0122
order S
0.2926::000118
0,2006:10.0044
0.1379:40.005
0.1084::0.005,
order
0.2285
0.1300:0000000
0.0898:10.00116
0.0688+10.0023
order 5 3
0.2058:10.0192
0.1123:10.0330
0.0712:0.0021
0.0585110.0030
(b) Predicting u.
Trasning size
5
10
order 0
0.6352-000673
0.452231000621
0.3570+10.0558
0.2737-10.0613
order <
0.3386+0.0250
0.2161 0088
0,1645++000114
0.1197+:00013
order 2
0.1716=0.000
0.0956=t00008
0.0680+0.0033
0.0642+0.0011
order < 3
0.2959+000088
0.1719::00213
0.1198+000018
0.0828+0.0055
Table 9: The relative Lg error with using different distribution in 6 network learning on Nonlinear Diffusion and Advection benchmark.
(a) testing f via & on different distribution of Nonlinear Diffusion benchmark.
Training size
10
30
d-network lest out of distribution
0.0092110.006 0.007110.0000 0002
0.0060110. 0.0003
0.0047-10.005
d-network test in distribution
0. 0.004410.0004
0.003410.0000 00004
O. 0.0028110.00 0.0002
0.0023:10.0003
(b) testing f via & on different distribution of Advection benchmark.
Training size
20
30
50
80
d-uetwork lest out of distribution 0.05561000077 0.0467+0.0065
0.04851100072
0.046110.0055 0.0397++0.000
0.0432-10.005p
g-network test in distribution
0.0423+40.0049
0.0365++000049
A Ablation study on the choice of derivatives. We further investigated the PPI learning performance with respect to the choice of derivatives used in our pseudo physics network. Specifically, tested PPI-FNO
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Table 10: The relative L2 error with using different distribution in pseudo- sesell--hysics-iiinee informed (PPI) learning on Nonlinear Difffusion and Advection benchmark.
(a) Predict u different distribution of Nonlinear Diffusion benchmark.
Training size
10
20
30
FNO
0. 1669910.0084
0. 0.00012
0.083710.0016
0.062610.0009
Ours
0.0229 + 0.0088
0.0179 + 0.0013
0.0135 + 0.0010
0.0103 + 0.0003
(b) Predict a on different distribution of Advection benchmark.
Training size
20
30
50
80
FNO
0.7039-
0. 0.7086+000076 0.6099 + 0.0227
0.6838--0.0258
0.6481
Ours
0.6175
0.0354
0.6231
0.0206
0.6471 + 0.0148
Table 11: Parameter counts for FNO and DONet with PPI variations across different problems. The training size is
30.
Parameter count
FNO
PPI-FNO (increase)
DONet
PPI-DONet (increase)
Darcy-Slow
1,188,353
1,229,476
(+3.46%)
2,084,704
2,125,827 (+1.97%)
Nonlinearddiffusoon
1,188,353
1,197,220
(+0.75%)
824,501
833,368 (+1.08%)
Fikonal
1,188,353
1,197,220
(+0.75%)
821,501
833,368 (+1.08%)
Poisson 1,188,353
1,197,220
(+0.75%)
821,501
833,368
(+1.08%)
Advection 1,188,353
1,197,220 (+0.75%)
210,101
218,968 (+4.22%%
on the Darcy-flow benchmark and varied the order of derivatives up to 0, 1, 2, and The performance is reported in Table [ We can see that although the accuracy of Φ with derivatives up to the third order is slightly better than with derivatives up to the second order, the best operator learning performance was still achieved using derivatives up to the second order (which was used in our evaluations). This might be because higher-order derivative information can cause overfitting in the pseudo physics network & to certain degree. Such higher-order information may not be critical to the actual mechanism of the physical system and can therefore impede the improvement of operator learning performance. Nevertheless, the framework itself not fundamentally limited to second-order terms: for higher-order PDEs, the input set of the surrogate ø-network can be straightforwardly extended to include third- or higher-order derivatives without modifying the core algorithm.
A Ablation study on the out-of-distribution (OOD) robustness. Beyond in-distribution accuracy, it is also important to understand how our method behaves under distribution shifts. We therefore investigate OOD robustness at two levels: (i) the pscudo physics network é, and (ii) the end-to-end PPI-FNO (comparison to FNO under identical OOD splits and training budgets). We construct OOD test sets by shifting the input distribution: for Advection we use squared exxoreentaa (SF) Gaussian kernels with test length scale 0.15 (training 0.25); for Nonlinear Diffusion we use test length scale 0.10 (training 0.20).
Rather than interpreting internal weights, we validate the learned surrogate by its input-outpen behavior. We compare ø's predicted dynamic responses against ground truth under challenging initial conditions for both in-distribution (ID) and OOD test fields, and we quantify alignment with physical principles. Across training sizes (Advection: N E (20, 30, 50, 80}; Nonlinear Diffusion: N E (5, 20, 30}), ó remains closcly aligned with the reference solutions on ID data and shows robust performance under moderate OOD shifts reported in Table g and heatmaps Figure 10 and Figure 11 These obscrvations indicate that the surrogate captures physically plausible behavior even as black box.
We then evaluate PPI-FNO and FNO the OOD splits with matched optimization settings. On Nonlinear Diffusion, both methods experience only mild degradation relative to ID, while PPI-FNO consistently outperforms FNO across N. On Advection, a larger length scale shift induces substantial degradation for all models; PPI-ENO remains competitive and typically stronger than FNO, but as expected for operator extrapolation- cannot fully recover accuracy far outside the training range. The performance is reported in Table 10 Overall, these results suggest that coupling with 6 improves sample efficiency and robustness under moderate shifts, but does not eliminate the fundamental difficulty of strong-range extrapolation.
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Relative L2 error
Relative L2 error
0.12
0.150
0.125
0.10
0.100
0.08
0.075
0.050
0.06
10
10
iteration
iteration
(a) PPI-FNO: Darcy Flow
(b) PPI-DONet: Nonlinear Diffusion
0.09
Relative L2 error
Relative L2 error
0.100
0.08
0.095
0.07
0.090
0.06
0.085
0.05
0.080
0
3
4
0
2
3
4
5
iteration
iteration
(c) PPI-FNO: Eikonal
(d) PPI-DONet: Eikonal
Figure 8: Predictive performance us. alternatingly fine-tuning iterations.
Relative L2 error
0.150
PPI-FNO
Relative L2 error
0.12
PPI-DONet DONet
FNO
0.125
0.10
0.100
0.075
0.08
0.050
10-1
100
10
10"
10a
102
À
A
(a) PPI-FNO: Darcy Flow
(b) PPI-DONet: Nonlinear Diffusion
Relative L2 error
0.09
PPI-FNO
Relative L2 error
PPI-DONet
0.08
FNO
0.10
DONet
0.07
0.09
0.08
0.05
100
10
102
100
10"
10a
A
A
(c) PPI-FNO: Eikonal
(d) PPI-DONet: Eikonal
Figure 9: Predictive performance weight A.
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Table 12: running time cost FNO and PPI-ENO on darcy-flow and Poisson benchmark.
(a) running time cost of standard FNO
Training size
5
10
20
30
darcy-flow
14
14
14
15
poisson
13
13
14
14
(b) running time cost of PPI-FNO
Training size
5
10
20
30
darcy-llow
984
1001
1036
1064
poisson
869
881
908
935
-20
-20
0.00
0.25
0.50
-25
25
-25
0.0
0,5
1,0
N-20
Ground truth (I))
d-pred (ID)
eerroor (ID)
Ground Truth (OOD)
pred (OOD)
e-eror (COD)
20
N=30
Ground truth (ID)
ppred (ID)
сerrro (ID)
Ground Truth (OOD)
A pred (OOD)
errro (COD)
-20
-20
0.0
0.7
-75
25
25
0.0
0,5
N=50
Ground-truth (ID)
d-pred (ID)
A eerror (ID)
pred (OOD)
e-error (COD)
-20
-20
0.0
0,1
-2
2>
25
25
0.00
0.25
0,50
N-80
Ground-truth (ID)
dxpred (ID)
(ID)
Ground Truth (OOD)
preeed (OOD)
(COD)
Figure 10: Advection heatmap (ID) OOD).
Ablation study on noisy data. The main experiments in this work focus on the challenge of limited data availabilily, i.e., small number of solution snapshots N, rather than low spatial resolution. The use of high-resolution (128x 128), noise-free data in our benchmarks was intentional: it provides a controlled and stringent testbed evaluate our method's ability learn generalizable physical model from few examples. If the data were coarse or noisy, would be difficult to disentangle the effect of data quality from the method's intrinsic performance. In addition, high-fidelity data is typically expensive and thus limited
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N=5
Ground-truth (ID)
+ pred (I))
errror (ID)
Ground Truth (COD)
pred (OOD)
eerror (OOD)
N=10
Ground truth (ID)
+ pred (ID)
crror (ID)
Ground Truth (COD)
+ pred (OOD)
-error (OOD)
0,4
N-20
Ground-truth (ID)
pred (I))
error (ID)
Ground Truth (COD)
pred (OOD)
$ error (OOD)
0,2
N-30
Ground-truth (ID)
A pred (ID)
error (ID)
Ground Truth (GOD)
$ pred (OOD)
+ error (OOD)
Figure 11: Nonlinear diffusion heatmap (ID) / OOD).
in quantity, whereas low-fidelity data is easy acquire and rarely scarce. Hence, our setting with few reflects the intended use case of the proposed framework.
high-fidelity
heleth evaluated the robustness of our framework to data sparsity and noise using the nonlinear diffusion benchmark. Each training input-output pair, originally sampled on a 128 x 128 noise- free grid, was randomly subsampled to 400 function values, to which we added 5% Gaussian noise. These noisy samples were then denoised and interpolated back to the 128 x 128 grid using Gaussian process interpolation (since FNO requires gridded data). We subsequently trained our PPI-NO model under this setting.
As shown in Table 13 our method consistently outperforms FNO across all training sizes, even in the presence of noise, though with the expected performance degradation compared to the clean-data experiments.
Memory and Time Cost. Our PPI-NO is more costly than standard NO since we need to train an additional "pseudo" physics network & along with the NO model. However, the network á is small as compared to the NO component - & simply a pixel-wise MLP coupled with one convolution filter, resulting in marginal increase memory cost. Table IT shows the paramcter count of FNO, DONet and their tito--pyyssssiionnmee versions. On average, PPI-FNO increases the number of parameters over FNO by 1. 1.29% while PPI-DONet over DONet by 1.89%. Tn terms of training time, the alternating
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Table 13: The relative Le error training with noise data in sheudo-physics-Bie informed (PPI) learning on Nonlinear Diffusion benchmark.
Trasning size
5
10
20
30
FNO
1..0111 0.1465+0.0062
0.12171000056 0.0848++0.0041 0.0593+00.002
0.089110.0061
0. 0.0606110.0020 0.0504+000006
Ours
optimization between the solution operator and the pseudo-physics surrogate is by design: it allows the two components to co-evolve, but introduces extra wall-clock time relative to single-network baselines. The cost grows approximately linearly with the number of alternations (and the per-iteration epoch budget). return, we observe consistent accuracy gains in the low-data regime. To make the computational overhead transparent, we report end-to-end wall-clock times on the Darcy flow and Poisson benchmarks in Table [2] all measurements are obtained on Linux workstation equipped with an NVIDIA GeForce RTX 4090 (24 GB).
6
Discussions and Limitations
6.1 More general PDE representation
Our framework can extend more general settings where operator learning involves mapping from coefficients and/or initial and/or boundary conditions to solutions.
Coefficient input. When serves coefficient function in the PDE, our e-network can be trained to learn equations of the form
L(x, (()
= 0,
where St denotes derivative operators, e.g., u, (ff(z) u(x)) Mix- The training loss be constructed analogously, ensuring that the dettwwrk'ss predictions approach zero valid training examples. In this setting, a key challenge is the identifiability of the learned pscudo-PDE. For example, the network can collapse to trivial zero output for any input. To address the challenge, we can employ more informative training objectives.
First, we can design the PDE loss so that the residual is close lo zero on correctly paired coefficient solution samples but large on randomly mismatched pairs drawn from the same coefficient distribution. For such correctly paired data (u, f), encourage zero residual, where the L T22( norm is approximated by averaging over randomly sampled locations æ in the domain Ω. In contrast, if we keep the same solution but pair it with a coefficient f randomly drawn from the same coefficient distribution, we expect that
.... (ttc:h/h...llle.......oooo......
0.
This suggests contrastive loss:
[ =| D1liI [u, B1li
where each (22) norm is approximated by averaging the squared residual over randomly sampled locations in the same domain Ω as the data. The first term enforces small residual on true cocfficient-solution pairs, while the second (erm explicitly penalizes zero (or very small) residuals on randomly mismatched pairs. Under this objective, constant-zero PDE network is no longer optimal: the model can strictly decrease ((0) by keeping the residual small on true pairs but making it large on random mismatches, which reduces degeneracy and improves identifiability in practice.
Second, we can also use likelihood- baased generative models such as PixelCNN (van den Oord et al. 2016). learning to maximize the probability density of coefficients and solutions at the sampled locations. The probability density is parameterized by the network é. This is complementary the residual-basee objective above.
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Initial conditions. For cases involving mappings from initial conditions to solutions, the PDE can be written as
du at - ,,,,,, t, x, ((u)),,,,
du can be estimated directly from data, and g can then be learned via When trajectory data are available, di the d-network. The mapping from f to target time 7 can be represented ur f So 9 di, where nunerical integration provides approximate PDE representation.
Boundary conditions. When f represents Dirichlet boundary conditions, the PDE surrogate can be modeled as
u(x) f(x) a(x) 0(1, u, Si
where f enforcos the boundary condition and axx) is composite distance function that vanishes the boundaries. Incorporating more general boundary conditions into the PDE surrogate learning, however, remains open research question.
6.2 Limitations and future work
We acknowlodge that our Pscudo Physics-Informed Neural Operator (PPI-NO) learning framework has several limitations.
First, our empirical validation concentrates standard operator-Learuing benchmarks and does not cover highly nonlinear, multi-scale PDEs (e.g., Navier Stokes, Euler). These regimes exhibit strong stiffness, intermittent structures, and scale separation that can destabilize training, alter inductive biases, and invalidate assumptions implicit in our datasets. As a result, the reported gains should not be interpreted as evidence of effectiveness in these harder settings. The methodology, loss balancing, and data priors used here may interact unfavorably with multi-scale dynamics, making generalization uncertain even when numerical resolution is increased.
Second, because the surrogate physics network ó is trained from limited examples, it overfit spurious regularities in the training distribution and internalize artifacts of discretization or preprocessing. Such overfitting may not be apparent from aggregate errors yet can surface as unstable extrapolation, biased fluxes, or residual patterns that mimic physics without reflecting underlying conservation or constitutive structure.
The risk grows when the training inputs have narrow support (e.g., restricted length scales or amplitudes), when measurement noise is structured, or when the training loss inadvertently rewards easy-to-fit nonphysical correlations. In these cases, & can act as a powerful but misleading prior on the operator learner, tightening training loss while harming reliability, like we showed in the ablation study the choice of derivatives.
Third, the framework validates & primarily by its input-output behavior, not by interpretability of its internal form. Consequently, apparent agreement with test data can be driven by correlations specific to the sampling protocol, interpolation scheme, or solver discretization, rather than by genuine physical invariants. This ambiguity amplified under PDE range shift: improvements in-distribution metrics do not guarantee presorvation of causal structure when inputs depart from the training manifold. Without direct identifiability of operators or invariants, it is difficult to determine when é captures a true governing relation versus a dataset-dependent proxy. Hence, conclusions about "learned physics" must be treated cautiously, especially when the ground-truth PDE is unknown.
Fourth, grid-based function values are strongly correlated, so the effective sample size for training the surrogate physics network is much smaller than the raw number of grid points. Although our design uses convolutional layers to aggregate local noighborhoods-theee quality relative to a pointwise MLP, this aggregation does not create independent evidence. In practice, it can lead to underestimated uncertainty and overconfident losses when many nearby pixels convey redundant.
exploiting spatial correlation and improving representation
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information. Moreover, the degree and structure of correlation resolution-dependent. fixed receptive field in pixels corresponds to different physical extents across meshes, and discretization choices (stencil width, padding, interpolation) imprint resolution-specife artifacts. Under resolution shifts or for PDEs with longer correlation lengths, anisotropy, or nonlocal couplings these effects can induce systematic bias in the learned surrogate, degrade cross-resolution generalization, and blur the distinction between genuine physical dependencies and grid-induced regularitics. Consequently, improvements observed at single resolution should not be interpreted as evidence of resolution invariance.
Finally, o trades interpretability for predictive flexibility. This blx character constrains the framework's role in scientific workflows that require explicit, human-readable laws, term-level attribution, or formal guarantees. Even when predictions are accurate, the absence of a transparent structure impedes scrutiny, reproducibility across laboratories with different preprocessing pipelines, and integration with analytical theory or established numerical methods. Moreover, the coupling between ø and the operator learner can obscure the source of errors: performance degradations may arise from Ø, the operator model, or their interaction, complicating diagnosis. As result, the method is better viewed a predictive surrogate within well-specified data regimes than tool for definitive law discovery.
7 Conclusion
We have presented a Pseudo Physics-Infrrmed Neural Operator (PPI-NO) learning framework. PPI-NO is based on our observation that a PDE system often characterized by a local combination of the solution and its derivatives. This property makes feasible learn an effective representation of the PDE system, even with limited data. While the physics delineated by PPI-NO might not precisely reflect true physical phenomena, our findings reveal that this method significantly enhances the efficiency of operator learning with limited data quantity.
However, our current method cannot learn PDE representations for which the input function f is the initial condition. In such cases, the mapping from the solution function to the initial condition requires a reversed integration over time, hence we cannot decouple the derivatives. To address this problem, we plan to explicitly model the temporal dependencies in the PDE representation, such as via the neural ODE design (Chen et al. 2018
8 Acknowledgements
This work has been supported by MURI AFOSR grant FA9550-20-110035 (Machine Learning and Physics- Based Modeling and Simulation), NSF CAREER Award 104-204,,95, NSF CSSI-2311685, DARPA SURGE HR001 1-25-C-0036 (Rapid certification of additive manufactured components using MT./AT and physics-based modeling), and NSF DMS-2529112 (Rapid Digital Twin model updating).
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Appendix
A Experimental Details
A.1 Darcy Flow
We considered stcady-state 2D Darcy Flow equation
Li ct al. 2020b
-V (V-((())Vu(t)) = f() x E (0, (),
u(x) = 0 0(0, 1)²,
(6)
where u(x) is the velocity of the flow, a(x) characterizes the conductivity of the media, and f(x) is the source function that can represent flow sources or sinks within the domain. In the experiment, our goal to predict the solution given the external source f. To this end, we fixed the conductivity a, which generated by first sampling Gauss random field a in the domain and then applying a thresholding rule: a(x) 4 if
a(x) < 0, otherwise a(x) 12. We then used another Gauss random field to generate samples of f. We followed (Li et al. 2020b) to solve the PDE using a second-order finite difference solver and collected the source and solution at a 128 128 grid.
A.2 Nonlinear Diffusion PDE
We next considered a nonlinear diffusion PDE,
0,u(2, t) = 10 u 6) + 10= + f(x,t), w(-1,t) = t) 0, u(x, 0) 0,
(7)
where (I, t) E [-1, 1]. Our objective is to predict the solution function given the source function f. We used the solver provided in QLu et al. 2022 and discretized both the imput and output functions at a 128 x 128 grid. The source ʃ was sampled from a Gaussian process with an isotropic square exponential (SE)
kernel for which the length scale was set to 0.2.
A.3 Eikonal Equation
Third, we employed the Eikonal equation, widely used in geometric optics and wave modeling. It describes given a wave source, the propagation of wavefront across the given media where the wave speed can vary at different locations. The equation is as follows,
[u(x)) f(x) x [0, 256] 256),
(8)
where u(x) is the travel time of the wavefront from the source to location x, denotes the Euclidean norm, and f(x) > the speed of the wave at x.
the experiment, we set the wave source at (0, 10). The goal is to predict the travel time given the heterogeneous wave speed f. We sampled an instance of f using the expression:
f(x) max max()(x), 1.0,
where g(-) is sampled from a Ganssian process using the isotropic SE kernel with length-scale 0.1. We employed the eikonalfm library httpt:://gtthbb. cctp///evinggansteee/eaoaa/////a httpe/////ee.kaaaaaaaee//eeaaete that implements the Fast Marching method Sethian (1999) to compute the solution
A.4 Poisson Equation
Fourth, we considered 2D Poisson Equation,
Au= f, in Ω 1]², ulap 0.
(9)
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where A is the Laplace operator. The solution is designed to take the form, ((()).22) オズ Σ j sin(ira1) cos(j and f(11, is correspondingly computed via the equation. To generate the datasct, set K and r 0.5, and independently sampled each element dij from uniform distribution [0, 1].
K وj=1 (i+
A.5 Advection Equation
Fifth, we considered a wave advection equation;
Ou du = f. E [0, te [0,1].
(10)
at
dr
The solution is represented by kernel regressor, u(x)= Σド wyk(x, 2j), and the source f is computed via the equation. To collect instances of (f, u), we used the square exponential (SE) kernel with length-scale 0.25. We randomly sampled the locations 2j from the domain and the weights wy from a standard normal distribution.
A.6 Fatigue Modeling
We considered predicting the SIF values along semi-ellipiic surface cracks on plates, as shown Fig 3 The SIF value can be viewed as a function of the angle ø E x], which decides the location of cach point on the crack surface. The geomelry parameters that characterize the crack shape and position were used as the imput, including a/c, a/t and c/b. In the operator learning framework, the input can be viewed as a function with three constant outputs. The dataset was produced via high-fidelity FE model under Mode tension (Merrell et al. 2024 Each data instance includes 128 samples of the SIF values drawn uniformly across the range of Φb
27


Reviews:
[{"review_id": "DiAgebVA56", "reviewer": "Reviewer_7jtP", "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": "In this paper, the authors introduce a Pseudo Physics-Informed Neural Operator (PPI-NO) framework --- the goal of operator learning is to approximate an operator that maps input parameters and/or functions to corresponding output functions.\n- For one example, such scenarios may arise in the context of solving partial differential equations.\n- Specifically, this paper aims to learn the solution operator of a PDE system, using a training dataset, which includes different instances of the PDE system sampled at a set of locations.\n\nThe concept of physics-informed neural operator (PINO) aims to embed physical laws, i.e., the governing equations, into the learning process.\n\nThis paper is motivated by the case when physics knowledge is absent, hence it is impossible to construct the physics loss term as in (PINO).\n\nAs a result, this paper proposes to design a neural network to approximate the general form of the PDE system. An $\\ell_2$ is used to reconstruct the parameters of the neural network. Next, the paper also proposes to leverage the pseudo-physics laws to enhance neural operator learning.\n\nFinally, this paper reports experiments that evaluate their proposed method on five commonly used benchmark operator learning problems in the literature, as well as an application of fatigue modeling.", "claims_and_evidence_yn": "No", "claims_explanation": "My understanding of this paper is that it proposes to use a neural network to approximate the outcomes of a PDE system, the mechanism for doing this is based on a literature called physics-informed neural operators. The main idea here is to approximate physical laws by closely tracking their outcomes in a surrogate model.\n\n- However, I find the connection to \"physical laws\" to be unconvincing --- I did not see either theoretical results or empirical evidences showing that the proposed modeling is capable of following a \"physical law.\"\n\n- The proposed algorithm is more like a Monte-Carlo approximation algorithm to approximate the instances from the input function space given enough samples. However, the paper does not state the sample size requirement for achieving this accurately. In the worst-case scenario, imagine learning a PDE whose derivative is specified by a piecewise linear function. The sample complexity would scale with the number of pieces, to say the least, as a lower bound. [Ref: Nearly-tight VC-dimension and Pseudodimension Bounds for Piecewise Linear Neural Networks. JMLR 2019]\n\nTherefore, the proposed algorithm in its full generality seemed hopeless but the paper does not specifiy the conditions or assumptions under which the algorithm is expected to work well.\n\n- The experiments including the detailed ablation studies are interesting. However, in table 2, it wasn't clear to me how to calibrate the reported numbers (are the reported results good enough for the proposed problem? many are in the range of 60% to 70%). As a result I was unable to evaluate the strengths of the empirical results. [Disclaimer: I have limited experience working in the area of \"physics-informed neural operators\" so my judgment may be biased here]\n\n- Finally, I think the paper could benefit from some additional editing and polishing as I noted plenty of typos and misformated content. For instance, the figures in Figure 8 and 9 should be downsized. Table 8 is placed right after the end of the conclusion.\n\nThe proposed algorithm could use a better explanation.", "audience_yn": "Yes", "audience_explanation": "The paper concerns learning neural networks to approximate the outcome of a PDE system. This may be related to applying ML to science broadly speaking but the connection is superficial and I think more examples and applications are required to further substantiate the contribution to the machine learning audience.", "requested_changes": "There is plenty of room for improvement in this paper. Here are several examples (but there are probably more than the items below that need to be addressed):\n\n- First, the paper requires a major revision to improve its formatting and its explanation of the main concepts.\n\n- Second, the paper needs to explain the conditions or assumptions under which their proposed algorithm will work more explicitly.\n\n- Third, I think the paper can add more examples and applications to substantiate why the proposed method is relevant and interesting to the machine learning audience.\n\n- Finally, it would be helpful to add a paragraph summarizing the contribution to the literature in the introduction.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "3SrjAiX2rx", "reviewer": "Reviewer_SQyJ", "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": "**Summary**\n\nThe paper introduces the Pseudo Physics-Informed Neural Operator (PPI-NO) framework, which couples a neural operator (e.g., FNO or DeepONet) with a learned “pseudo-physics” surrogate PDE network. The surrogate network takes solution values and derivative approximations as inputs and is trained jointly with the operator via alternating optimization. The goal is to improve operator learning in scenarios where governing physics are unknown and labeled data are scarce. The paper presents extensive experiments on benchmark operator learning problems and a fatigue modeling application, along with ablation studies.\n\n**Key strengths** include:\n- **(1)** The problem of enhancing neural operators when physical laws are unavailable and labeled data are scarce is an important and underexplored research direction.\n-  **(2)** The core methodology is clearly presented, and the alternating training scheme is well motivated.\n-  **(3)** The paper provides extensive numerical examples, including ablations, which is commendable.\n\n**Key weaknesses** include:\n- **(1) Weak motivation**: The central idea is to improve neural operator training by introducing a surrogate “pseudo-physics” network that acts as a learned PDE. However, I find the motivation weak. There already exist advanced system identification methods capable of accurately recovering governing equations from limited data. A more natural and efficient pipeline would be: **(i)** use system identification to learn approximate governing laws from scarce labeled data (a relatively fast process),  **(ii)** generate abundant synthetic data via the identified system, **(iii)** train a neural operator on this augmented dataset (very fast process). This **\"system-identification-then-neural-operator”** approach is conceptually simpler, computationally cheaper, and well aligned with existing literature. Unless the authors can demonstrate that PPI-NO achieves substantial accuracy or efficiency gains over such a pipeline, the proposed coupling strategy appears unnecessarily expensive and poorly motivated.\n-  **(2) Problem specificity:** The proposed framework is narrowly applicable. It only learns mappings from source terms $f$ to solutions $u$ for PDEs with at most second-order derivatives, since the surrogate network $\\phi$ is designed to take inputs of $u$ and its first- and second-order derivatives. This excludes broader classes of problems, such as mappings from initial conditions or coefficients to solutions, or PDEs involving higher-order derivatives.\n- **(3) Lack of interpretability or diagnostics for the learned pseudo-physics**: The paper recognizes that the learned physics network is a flexible black-box and tradeoffs interpretability for accuracy, but does not provide in-depth analysis or visual diagnostics of what types of surrogate physics are actually learned, nor how they relate to the true underlying physics when available. This could limit the practical adoption of the framework in some scientific domains.\n- **(4) Unfair comparisons:** The reported advantages of PPI-NO are not convincing due to unfair experimental design. As shown in Eq. (5), PPI-NO effectively leverages additional training data from the surrogate $\\phi$ ($N’$ low-fidelity pairs) beyond the $N$ high-fidelity pairs used by baselines such as FNO and DeepONet. This extra supervision is a key reason for PPI-NO’s superior performance. For fairness, either: (i) all methods should be trained with the same total amount of data (e.g., $N$ high-fidelity pairs), or (ii) baselines should be provided with analogous low-fidelity data (e.g., $N'$ low-fidelity pairs via a system-identification step). Without such adjustments, the current comparisons overstate the advantages of the proposed method.\n- **(5) Idealized data assumptions:** The experiments assume access to high-fidelity ($128\\times 128$ regular grids), noise-free data, which contradicts the paper’s motivating scenario. In realistic settings, data are more likely to be low-fidelity (e.g., coarse grids, sparse samples) and noisy. The framework’s robustness to these challenges is not tested.\n- **(6) Missing out-of-distribution evaluation:** A key promise of operator learning is generalization across problem settings. However, the paper only tests in-distribution performance. There are no experiments on out-of-distribution generalization (e.g., extrapolating to unseen input ranges). Without this, the claims about efficiency and robustness remain incomplete.\n- **(7) Missing scalability and computational cost:** Although memory costs are briefly discussed, the increased training complexity and wall-clock time due to alternating training of two networks is not analyzed quantitatively; such reporting would enhance transparency for practical use.\n- **(8) Incomplete discussion of limitations**: The paper does not adequately discuss the potential drawbacks of introducing a black-box pseudo-physics network, such as risks of overfitting, spurious inductive biases, or reduced trustworthiness in scientific discovery. A more balanced discussion would help position the work more realistically.", "claims_and_evidence_yn": "No", "claims_explanation": "The paper reports improved performance of the proposed PPI-NO framework over baseline methods, but the comparisons are not fully fair: \n- (1) PPI-NO leverages additional training data (via the pseudo-physics surrogate) that the baselines do not, making it difficult to isolate the benefit of the framework itself. \n- (2) Moreover, the motivation that PPI-NO is more effective than simpler alternatives, such as a “system-identification-then-neural-operator” pipeline, is not substantiated through experiments. \n- (3) The method’s applicability is also restricted to PDEs with at most second-order derivatives, yet this limitation is not clearly acknowledged. \n- (4) Finally, the evaluations are carried out only in idealized high-fidelity, noise-free settings, with no evidence provided for robustness to noisy data, low-fidelity observations, or out-of-distribution scenarios. \n\nTogether, these gaps reduce the strength and clarity of the evidence supporting the paper’s central claims.", "audience_yn": "Yes", "audience_explanation": "The topic of enhancing neural operator performance under limited labeled data and unknown physics is timely and aligns with the interests of the TMLR community, especially those working at the intersection of machine learning and scientific computing. The proposed PPI-NO framework introduces an unconventional idea of coupling a neural operator with a learned pseudo-physics surrogate, which could stimulate discussion. However, the current limitations—restricted applicability to second-order PDEs, reliance on idealized high-fidelity data, and lack of fair baselines—significantly weaken its practical impact. While some readers may find the methodology and experimental setup interesting, the broader audience may question its novelty, fairness, and real-world relevance in its present form.", "requested_changes": "**Critical Changes (required for acceptance)**:\n- (1) **Clarify and justify the core motivation**. The authors should provide a stronger justification for why coupling a neural operator with a pseudo-physics network is necessary and beneficial, particularly in comparison to a simpler “system-identification-then-neural-operator” pipeline. A direct and fair comparison in both accuracy and efficiency would be essential.\n- (2) **Fair baselines.** Ensure all comparisons are fair by using the same amount of training data for all methods. For example, if the proposed framework leverages additional pseudo-labeled data from the learned physics model, then the same number of data points should also be provided to FNO, DeepONet, or other baselines.\n- (3)\t**Scope of applicability.** The current method is limited to PDEs with maximum second-order derivatives and only applies to mapping source terms $f$  to $u$. The paper should clearly state this limitation and, ideally, explore extensions or discuss how the framework could handle broader classes of PDEs (e.g., systems with coefficients, higher-order derivatives, or initial-value problems).\n- (4) **More realistic experiments.** Include experiments where training data is (i) available only on coarse or randomly sparse grids, and (ii) noisy or corrupted, as these reflect the very scenarios of scarce and imperfect data that motivate the paper.\n- (5) **Out-of-distribution generalization.** Provide experiments to assess whether the proposed approach generalizes to inputs outside the training distribution, which is critical for operator learning tasks.\n- (6) **Scalability and cost analysis.** Quantitatively analyze the runtime and memory overhead introduced by alternating optimization between two networks. This information is important for understanding the method’s practicality.\n\n**Suggested Changes (to strengthen the work)**:\n- (1) **Interpretability of pseudo-physics.** Provide illustrative examples or visual diagnostics of what the pseudo-physics network actually learns. This would help readers understand whether it captures meaningful surrogate physics or simply acts as a flexible regularizer.\n- (2) **Sensitivity analysis.** Investigate the robustness of the method to variations in the architecture and parameterization of the pseudo-physics network (e.g., depth, width, inclusion/exclusion of derivative terms).\n- (3) **Discussion of limitations.** Add a clear section outlining when the proposed method is most suitable and when it may not be appropriate (e.g., black-box surrogate overfitting, reduced interpretability).\n- (4) **Clarify data usage.** On page 3, clarify the statement regarding decomposing a $128 \\times 128$ input into 16,384 points. This does not increase information content, and a clearer explanation is needed.", "broader_impact_concerns": "I do not see major ethical or societal risks tied to this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "BkojlHzFf7", "reviewer": "Reviewer_oRPB", "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 the Pseudo Physics-Informed Neural Operator a novel framework that enhances neural operator learning in data-scarce regimes. The method constructs a surrogate PDE system using finite-difference derivatives and learnable convolutional filters, which is then coupled with existing neural operator architectures (e.g., FNO and DeepONet) through an alternating training scheme.\n\nSome of the contributions include:\n\n- Demonstrating that surrogate, data-driven “pseudo-physics” can serve as an effective regularizer for operator learning without requiring full knowledge of governing equations.\n\n- Comprehensive experiments on benchmark PDE problems (Darcy flow, nonlinear diffusion, Eikonal, Poisson, Advection) as well as a challenging fatigue modeling task, showing substantial performance improvements, particularly with very limited training data.\n\n- Ablation studies that carefully assess the role of the pseudo-physics network, derivative orders, convolutional layers, loss weights, and training strategies.\n\n----\n\nStrengths:\n\n- Novel and creative approach to bridging physics-informed and purely data-driven operator learning.\n\n- Strong empirical validation across diverse PDEs and a real-world fatigue modeling application.\n\n- Clear demonstration that the framework is especially effective in low-data regimes.\n\n- Well-written and thorough, with multiple ablations that strengthen the claims.\n\nWeaknesses:\n\n- Limited discussion on how discovered PDEs generalize (especially out-of-distribution generalization), particularly when spurious or nonlinear terms are learned.\n\n- Open questions remain about the applicability to highly nonlinear, multi-scale systems (e.g., Navier–Stokes, Euler equations), where physics-informed methods are known to struggle.\n\n- Some assumptions on data independence (e.g., treating grid points as separate samples) could be misleading", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors provide extensive experimental results across five benchmark PDE problems and one applied fatigue modeling scenario. Performance comparisons against FNO, DeepONet, and PINO baselines are fair and transparent. Ablation studies on architecture, hyperparameters, and derivative order further strengthen the evidence. While certain theoretical aspects (e.g., robustness to incorrect pseudo-physics) are not fully explored, the empirical evidence convincingly supports the main claims of improved data efficiency and predictive accuracy.", "audience_yn": "Yes", "audience_explanation": "The work is highly relevant to the TMLR community, particularly researchers in scientific machine learning, operator learning, and data-efficient modeling. The introduction of pseudo-physics-informed learning provides both a conceptual contribution and a practical framework that could inspire further research, especially in domains where ground-truth physics is unknown or partially understood. The unusual and useful fatigue modeling benchmark also broadens its potential impact.", "requested_changes": "Critical: Clarify the limitations of the approach in cases where the discovered PDE is incorrect, especially regarding its effect on generalization (both in-distribution and out-of-distribution). While the paper does acknowledge that the learned pseudo-physics may not exactly reflect the true governing equations and presents empirical evidence that operator learning is relatively robust to inaccuracies in the pseudo-physics network, the discussion remains limited. The paper does not fully analyze potential failure cases, for example, when spurious nonlinear terms are introduced, or when the learned PDE fundamentally misrepresents the underlying system. In such cases, one might expect both in-distribution and out-of-distribution generalization to suffer. A more explicit discussion of these limitations would strengthen the paper and help readers understand the boundary conditions under which pseudo-physics-informed methods remain beneficial.\n\nCritical: Provide some discussion of applicability to nonlinear, multi-scale PDEs (e.g., Navier–Stokes, Euler). Even if experiments are not feasible, a short argument about feasibility and challenges would strengthen the paper.\n\nOptional: Elaborate on whether and how parameters, source terms, or boundary/initial conditions could be incorporated into the pseudo-PDE network.\n\nOptional: Revisit the discussion around data independence, while the framework treats all grid points as separate samples, in many cases solutions are parametrized, and thus this assumption may overstate the effective data richness.", "broader_impact_concerns": "No major ethical concerns are apparent.", "additional_comments": "/", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)