Boundary value inverse problems aim to recover the internal structure or distribution of multiple media inside an object (2D reconstruction) based on only the data available on the boundary (1D signal input), which arise from many imaging techniques, e.g., electrical impedance tomography (EIT) (Holder, 2004), diffuse optical tomography (DOT) (Culver et al., 2003), magnetic induction tomography (MIT) (Griffiths et al., 1999). Not needing any internal data renders these techniques generally non-invasive, safe, cheap, and thus quite suitable for monitoring applications.
Let be a nonlinear operator associated a physical model governed by certain partial differential equations (PDE) on a bounded in . For the measurement on the boundary , and the coefficient of the underlying PDEs to be recovered, the forward PDE model is
where is a noise following certain distribution. Comparing with the classic linear inverse problems
in computer vision and signal processing, e.g.(Marroquin et al., 1987), to recover a signal from measurement with noise , the fundamental difference is that is highly nonlinear, and the Hilbert spaces and are usually infinite dimensional. In many situations, even though the inverse operator is theoretically well-defined, as the infinite dimensional is practically approximated by a finite dimensional (sub)space , the problem of seeking the approximated operator is usually highly ill-posed (not having a well-defined unique output) and poses great challenges to the reconstruction algorithms.
2 Background, related work, and contributions
Classical iterative methods.
There are in general two types of methodology to solve inverse problems. The first one is a large family of iterative or optimization-based methods (Dobson & Santosa, 1994; Martin & Idier, 1997; Chan & Tai, 2003; Vauhkonen et al., 1999; Guo et al., 2018; Rondi & Santosa, 2001; Chen et al., 2020). One usually looks for the desired by solving a minimization problem
where is a regularization term to alleviate the ill-posedness, and its design plays a critical role for a successful reconstruction (Tarvainen et al., 2008; Tehrani et al., 2012; Wang et al., 2012). For almost all these iterative methods, due to the ill-posedness, the computation generally takes numerous iterations to converge, and the reconstruction is highly sensitive to noise. Besides, needs to be evaluated at each iteration, which is itself expensive as it requires solving forward PDE models.
The second methodology is to develop a well-defined mapping parametrized with , empirically constructed to approximate the inverse map itself:
This method is referred to as non-iterative or direct methods in the literature. Distinguished from iterative approaches, direct methods are in general highly problem-specific, as they are designed on an ad hoc basis based on the mathematical structure of the various inverse operators. For instances, methods in EIT and DOT include factorization methods (Kirsch & Grinberg, 2007; Azzouz et al., 2007; Brühl, 2001; Hanke & Brühl, 2003), MUSIC-type algorithms (Cheney, 2001; Ammari & Kang, 2004, 2007; Lee et al., 2011), and the D-bar methods (Knudsen et al., 2007, 2009) based on a Fredholm integral equation (Nachman, 1996), among which are the direct sampling methods (DSM) (Chow et al., 2014, 2015; Kang et al., 2018; Ito et al., 2013; Ji et al., 2019; Harris & Kleefeld, 2019; Ahn et al., 2020) being our focus in this work. These methods generally have a closed-form expression of to approximate , and the parameters have clear mathematical meaning. For each fixed , this procedure is usually much more stable than iterative approaches with respect to the input data. Furthermore, the evaluation for each given is distinctly fast as no training optimization is needed anymore. However, such a simple closed form of admitting efficient execution may not be available in practice since some mathematical assumptions and derivation may not hold.
Boundary value inverse problems.
For most cases of boundary value inverse problems in 2D, one key difference, e.g., with an image inverse problem, is that data are only available on 1D manifolds, which are used to reconstruct 2D targets. The boundary data themselves generally involve certain input-output structure, which adds more complexity. In Adler & Guardo (1994); Fernández-Fuentes et al. (2018); Feng et al. (2018) boundary measurements are collected and directly input into feedforward fully connected networks. As the data reside on different manifolds, special treatments are made to the input data, such as employing pre-reconstruction stages to generate rough 2D input to CNNs (Ben Yedder et al., 2018; Ren et al., 2020).
Operator learning and inverse problems.
Solving an inverse problem is essentially to approximate the inverse operator
but based on finitely many data. The emerging deep learning (DL) technique makes it possible to directly emulate operators. The concept resembles greatly the aforementioned classical direct methodology. But operator learners by DNNs are generally considered as black boxes. A natural question to ask is whether and how existing innovative neural architectures, usually designed for other purposes, can be modified conforming with the mathematical nature of the underlying problem, which ultimately leads to structure-conforming DNNs. Inspired by classical direct methods, we try to improve the DNN reconstruction pipeline from the architectural perspective. In this regard, the proposed study provides a positive example to a hopefully definitive answer to this question, which bridges deep learning and conventional tasks in physical sciences.
Operator learning has become an active research field for inverse problems in recent years, especially related to image reconstruction where CNN plays a central role, see e.g, Kłosowski & Rymarczyk (2017); Nguyen et al. (2018); Tan et al. (2018); Jin et al. (2017); Kang et al. (2017); Barbastathis et al. (2019); Latif et al. (2019); Zhu et al. (2018); Chen et al. (2021); Coxson et al. (2022). Some efforts have been made to couple classical reconstruction methods and CNN. Notable examples include Hamilton et al. (2019); Hamilton & Hauptmann (2018), where a CNN post-processes images obtained by the classical D-bar methods; Fan et al. (2019); Fan & Ying (2020) develop BCR-Net to mimic pseudo-differential operators appearing in many inverse problems; a deep direct sampling method is proposed in Guo & Jiang (2020); Jiang et al. (2021)
that learns local convolutional kernels mimicking the gradient operator of DSM. Another example is radial basis function neural networks seen inHrabuska et al. (2018); Michalikova et al. (2014); Wang et al. (2021a). Nevertheless, convolutions in CNNs use locally supported kernel whose receptive field involves only a small neighborhood of a pixel, thus layer-wise speaking, CNN does not align well with the non-local nature of inverse problems. More recently, the learning of PDE-related forward problems using global kernel has gained attraction, most notably the Fourier Neural Operator (FNO) (Li et al., 2021)
. FNO takes the unique advantage of the low-rank nature of certain problems, and proposes to train a local kernel in the frequency domain yet global in the spatial-temporal domain, mimicking the solution’s kernel integral form. Others include DeepONets(Lu et al., 2021; Wang et al., 2021b; Jin et al., 2022) and Transformers (Kissas et al., 2022; Li et al., 2022; Cao, 2021).
Related studies on Transformers.
The attention mechanism-based models have become the state of the art in many areas since Vaswani et al. (2017), such as language tasks and computer vision e.g., Ma et al. (2021); Dosovitskiy et al. (2021). One of the most important and attractive aspects of the attention mechanism is its unparalleled capability to model long-range interactions (Tay et al., 2021) through many efficient variants see e.g., Katharopoulos et al. (2020); Choromanski et al. (2021); Nguyen et al. (2021). The relation of the attention with kernel learning is firstly studied in (Tsai et al., 2019) and later connected with random feature (Peng et al., 2021). Connecting the non-PDE-based integral kernels and the attention mechanism has been seen in Hutchinson et al. (2021); Guibas et al. (2022). Among inverse problems, Transformers have been applied in medical imaging applications including segmentation problems (Zhou et al., 2021; Hatamizadeh et al., 2022; Petit et al., 2021), X-Ray (Tanzi et al., 2022), magnetic resonance imaging (MRI) (He et al., 2022), ultrasound (Perera et al., 2021), optical coherence tomography (OCT) (Song et al., 2021), etc. To our best knowledge, there is no work in the literature that connects the attention mechanism in Transformer and the mathematical structure of PDE-based inverse problems.
A structure-conforming network architecture. Inspired by the classical DSM, we decompose the inverse operator into a harmonic extension and an integral operator with learnable non-local kernels that has an attention-like structure. Additionally, the attention architecture is reinterpreted through a Fredholm integral operator to rationalize the application of the Transformer to the boundary value inverse problem.
Theoretical and experimental justification for the advantage of Transformer. We have proved that, in Transformeris, a modified attention is able to represent target functions exhibiting higher frequency natures from lower frequency input features. In the experiments, a comparative study further demonstrates a favorable match between the Transformer and the problem structure.
3 Interplay between mathematics and neural architectures
In this section, we use EIT, a classical boundary value inverse problem, as a prominent example to walk through that the triple tensor product in the attention mechanism matches extremely well with representing a solution in the inverse operator theory of EIT. The proposed method is motivated by and in pursuit to provide examples to the answers of the following questions
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What is an appropriate finite dimensional data format of as inputs to the neural network?
Is there a suitable neural network in (3) taking advantage of the mathematical structure?
3.1 From EIT to Operator Learning
The forward model of EIT is given by the following second order elliptic partial differential equation
Given the domain , the goal is to identify the unknown inclusion buried in . The value and are two (approximately) known constants discontinuous across the boundary . Without loss of generality, we assume . Then, the coefficient of (4
) to be recovered can be described by a characteristic functiondefined for each point , and is for and for , or equivalently . Then,
is a vector of values at the grid points approximating(see Figure 2 in Appendix B for an example discretization).
In the application of EIT, the boundary measurement space contains information of how the electric potential behaves on the boundary. By exerting a current on the boundary, solving (4) for a unique with this specific boundary information (Neumann boundary condition) reveals ’s behavior on the whole domain . However, what can be measured is only the voltages on the boundary. This procedure is called Neumann-to-Dirichlet (NtD) mapping:
The - duality for Sobolev spaces here is for formality, and we refer the readers to Appendix A for more detailed descriptions of function spaces. One measurement is a single data pair , and all the measurements form this infinite-dimensional space:
If the NtD mapping is known, the theoretical uniqueness of can be established (Brühl, 2001; Hanke & Brühl, 2003; Astala & Päivärinta, 2006; Nachman, 1996). Mathematically, it means if one is able to measure all the current-to-voltage pairs in , where form a set of basis functions of the Hilbert space , then , which is essentially , is well-defined admitting explicit format (Brühl, 2001).
We shall illustrate how the mathematical setup above is realized in a discrete level. For each fixed , the NtD mapping in (5) can be expressed as
where and are tensors containing boundary data at grid points on the boundary. If is known, then (6) is obtained through a typical PDE solver. For unknown , using a significantly large number of pairs of - can obtain a huge matrix approximating to on a reasonably fine grid. The mechanism here essentially results in a tensor2tensor mapping/operator from to the images of . In this case, the operator can be learned through a large number of - data pairs. In particular, the BCR-Net (Fan & Ying, 2020) is a DNN approximation falls into this category.
However, as infinity or large number of boundary data pairs is not attainable in practice, the problem of more practical interests is to use only a few data pairs for reconstruction with being very small not enough to get any reasonably good approximation of in (6). Let be the subspace spanned by . Here, the highly nonlinear forward operator for finitely many pairs is still trivially well-defined. However, it becomes a long-standing open theoretical problem whether its inverse is well-defined. Namely, it is unclear if the inclusion can be exactly identified by finite measurement on boundary.
As we assume , the operator
is positive definite, and it has eigenvalueswith (Cheng et al., 1989). The following theorem justifies the practice of approximating when is large enough. Roughly speaking, as is well defined, when , we can define an to approximate .
Suppose that the 1D boundary data is the eigenfunction of
is the eigenfunction ofcorresponding to the -th eigenvalue , and let the 2D data functions be obtained by solving
where . Define . Then, for any , there exists a sufficiently large such that .
The proof of Theorem 1 can be found in Appendix C. Nevertheless, the constructional proof of the existence of still relies on the entire NtD mapping , which again resorts to infiniteness, thus inaccessible in real applications. Another valuable corollary of Theorem 1 is to justify using the data functions as the input to neural networks, see Section 3.3 for more details.
Operator learning problems for EIT.
Many works have made efforts to derive a well-defined operator that estimatesinstead of an exact or accurate reconstruction (Brühl, 2001; Chow et al., 2014; Dunlop & Stuart, 2015) for even . Such effort could benefit modern operator learning methods. To this end, we first introduce several attainable approximations of infinite dimensional spaces by finite dimensional counterparts for the proposed method.
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Spatial discretization. Let be a mesh of with the mesh spacing and let be the set of grid points to represent the 2D discretizations for continuous signals. Then a function defined almost everywhere in can be approximated by a vector .
Sampling of coefficient or equivalently . samples of with different shapes and locations following certain distribution are used approximate . is usually large enough to represent field applications of interest. For the -th sample of , we let be a discretization of on . Accordingly, is a data function associated with .
Sampling of NtD mapping. The most highlighted approximation is, for each sample of , there are input-output data pairs to sample the NtD mapping , which are used to generate channels in an input image. Despite a “sufficiently large” needed in Theorem 1, in practice, can be chosen to be very small () to yield satisfactory results.
Our task is to find a parameterized mapping to approximate by minimizing
The parameters will affect the finite approximation to infiniteness in the following way: determines the resolution to approximate ; affects the representativity of the training data set; decides how much of a finite portion of the infinite spectral information of can be accessed.
3.2 From channels in attention to basis in integral transform
In this subsection, a modified attention mechanism is proposed as the basic block in the tensor2tensor-type mapping introduced in the next two subsections. Its reformulation conforms with one of the most used tools in applied mathematics: the integral transform. Solutions to many physical problems have this form, as it aggregates the interactions of a class of functions at different locations to characterize how the operator mapping behaves between infinite-dimensional spaces. More interestingly, in most applications, the interaction (kernel) does not have any explicit form, which meshes well with DL methodology philosophically. In fact, this is precisely the situation of the considered problem.
For simplicity, let the input of an encoder attention block be with channels, then the query , key , value are generated by three learnable projection matrices : , , . Here is the number of expanded channels for the latent interactions. A modified dot-product attention is proposed as follows:
where and are two learnable normalizations. Different from Nguyen & Salazar (2019); Xiong et al. (2020), this pre-inner-product normalization is applied right before the matrix multiplication of query and key, which takes inspiration from the normalization in the index map kernel integral (13) and (22), see also Boyd (2001) where the normalization for orthogonal bases essentially uses the (pseudo)inverse of the Gram matrices. In practice, its cheap alternative is chosen to be layer normalization (Ba et al., 2016)Ioffe & Szegedy, 2015). is a mesh-based weight such that the summation becomes an approximation to an integral quadrature.
To elaborate these rationales, the -th column of the -th row of is , in which the -th row and . Thus, applying this to every column , attention (9) becomes a basis expansion representation for
Here, contains the coefficients for the linear combination of . This set form the ’s row space, and it further forms each row of the output by multiplying with . in (10) stands for the attention kernel, which aggregates the pixel-wise feature maps to measure how the input latent representations interact. Moreover, the latent representation in an encoder layer is spanned by the row space of , and is being nonlinearly updated layer-wise.
If we further assume that there exist a set of feature maps for normalized query and key, as well as value, e.g., see Choromanski et al. (2021). For , the feature map that maps , i.e., , then a kernel can be defined by
Now the discrete kernel with vectorial input is rewritten to an integral kernel , thus the dot-product attention is expressed as a nonlinear integral transform for the -th channel:
After plugging in certain minimization such as (8
), the backpropagation updates weights, which further lead a new set of latent representations. This procedure can be viewed as an iterative method to update the basis residing in each channel by solving the Fredholm integral equation of the first kind in (12).
To connect attention with inverse problems, the multiplicative structure in a kernel integral form for attention (12) is particularly useful. This falls into the category of Pincherle-Goursat (degenerate) kernels (Kress, 1999, Chapter 11), and its approximability depends on the number of the expanded channels . decides number of learned basis functions in expansion (10
), a subset of which eventually is used to form a set of basis. Here we show the following theorem, heuristically it says that: given enough but finite channels of latent representations, the attention kernel integral is able to “bootstrap” in frequency domain, that is, generating an output representation with higher frequencies than the input. Similar approximation result is impossible for CNN if opting for the usual framelet/wavelet interpretation(Ye et al., 2018). The full proof with a more rigorous setting is in Appendix D.
Theorem 2 (Frequency bootstrapping).
Suppose there exists a channel in such that for some , the current finite-channel sum kernel approximates a non-separable kernel (e.g., Fourier-type ) to an error of under certain norm . Then, there exists a set of such that certain channel in the output of (10) approximates , with an error of under the same norm.
The considered inverse problem is essentially to recover high-frequency eigenpairs of based on the low-frequency data, see e.g. Figure 1. together with all its spectral information can be determined by the recovered inclusion shape, thus the existence result in Theorem 2 justifies the advantages of the attention mechanism for the considered problem.
3.3 From harmonic extension to tensor-to-tensor
To establish the connection between the problem of interest with the attention used in the Transformers, we begin with the case of a single measurement, i.e., . With this setting, it is possible to derive an explicit and simple formula to approximate inspired by direct sampling methods (DSM) (Chow et al., 2014, 2015; Kang et al., 2018; Ito et al., 2013; Ji et al., 2019; Harris & Kleefeld, 2019; Ahn et al., 2020) for a larger family of inverse problems. For instance, an approximation modified from the one in Chow et al. (2014) reads as
where , and is the boundary data measuring the difference of NtD mappings. In particular, is the solution to (7) with but with certain noise on the boundary, i.e., is replaced by , which is called the harmonic extension. The vector function is called probing direction and can be chosen empirically as . Lastly, is the solutions to:
where restricted on boundary is equipped with semi-norm, and is function associated with the point . Both and can be computed effectively by traditional fast PDE solvers, such as finite difference or finite element methods based on in Section 3.1. Essentially, (13) suggests that the derivative of along a direction scaled by a quantity approximates , where and are based on empirical choices. Indeed, the reconstruction accuracy is much limited by a single measurement, the nonparametric ansatz, and empirical choices. These restrictions give room for DL methodology.
Despite being restrictive, the formulation in (13) offers inspiration to give a potential answer to (Q1): the harmonic extension of can be used as the input to a tensor2tensor-type DNN. Constructing harmonic extension (2D features) from boundary data (1D signal input with limited embedding dimensions) can contribute to the desired high-quality reconstruction. First, harmonic functions are highly smooth away from the boundary, of which the solution automatically smooths out the noise on the boundary due to PDE theory (Gilbarg & Trudinger, 2001, Chapter 8), and thus enable the reconstruction being highly robust with respect to the noise (e.g., see Figure 1). Second, in terms of using certain backbone network to generate features for downstream tasks, harmonic extensions can be understood as a problem-specific way to design higher dimensional feature maps (Álvarez et al., 2012), which renders samples more separable in a higher dimensional data manifold than the one with merely boundary data.
The information of is deeply hidden in . As shown in Figure 1, one cannot observe any pattern of directly from , for more examples see Appendix B. It is different from and much more difficult than the inverse problems studied in (Bhattacharya et al., 2021; Khoo et al., 2021) that aim to reconstruct 2D targets from the much more informative 2D internal data of .
As shown in Figure 1, can be nicely decoupled into a composition of a parametrized neural network operator and a non-learnable harmonic extension feature map , i.e., .
3.4 From index map integral to Transformer
In this last subsection, the probing direction , the inner product , and the norm in (13) are used as ingredients to form certain non-local learnable kernel integration. This non-localness is a fundamental trait for many inverse problems, in that depends on the entire data function. Then, the discretization of the modified index function is shown to match the multiplicative structure of the modified attention mechanism in (9).
In the forthcoming derivations, and a self-adjoint positive definite linear operator , are shown to yield the emblematic -- structure of attention. To this end, we make the following modifications and assumptions to the original index map in (13).
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Similarly, the probing direction is reasonably assumed to have a global dependence on
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: the vector that interpolatesat the grid points , .
: the discrete Laplacian on of a finite element/finite difference discretization of coupled with the Neumann boundary condition and the zero integral normalization condition.
: the matrix that projects a vector defined at to one defined at the nodes on the boundary .
We shall discretize the variable by grid points in (15) and obtain an approximation to the integral:
We then consider (16) and focus on one component of . With a suitable quadrature rule to compute the integral, it can be rewritten as
Next, we proceed to express by discretizing the variational form in (14) by a linear finite element method. Let be the set of the basis functions, and be the vector approximating for each fixed . Denote as the vector approximating and as its restriction on the boundary nodes. Then, the finite element discretization yields the linear systems:
Note that the self-adjoint positive definite operator can be parameterized by a symmetric positive definite (SPD) matrix denoted by . We can approximate as
Now, using the notation from Section 3.2, we denote the learnable kernel matrices and an input vector: for , and
Then, we are able to rewrite (22) as
where is a constant, and both and are taken element-wise. Here, we may define , , and as the values, keys and query. We can see that, the right matrix multiplications in (9) are low-rank approximations to the ones above in the attention mechanism. Hence, based on (24), essentially we need to find a function resulting in a vector approximation to the true characteristic function
Thus, the expression in (24) and (25) reveal that a Transformer is able to capture the classical formula in (13) equipped with non-local learnable kernels. Moreover, when there are data pairs, the data functions are generated by computing their harmonic extensions as in (7). Then, each harmonic extension is then treated as a channel of the input .
The derivation above motivates that the attention mechanism nicely fits the underlying mathematical structure. In summary, we propose to use a Transformer-based deep direct sampling method, and subsequently show that it has significantly better performance than the widely used CNN-based U-Net (Guo & Jiang, 2020) in linear inverse problems and modern operator learner such as Fourier Neural Operator (Li et al., 2021). In this regard, we provide a potential answer to the question (Q2); namely, the attention-based Transformer is better suited as it conforms more with the underlying mathematical structure, and both enjoy a global kernel formulation using the input data, which match the long range dependence nature of inverse problems.
In this section we present some experimental results to show the quality of the reconstruction using a single channel () of the 2D harmonic extension feature from the 1D signal input. There are two baseline models to compare: one is the CNN-based U-Net (Ronneberger et al., 2015), the other is the state-of-the-art operator learner Fourier Neural Operator (FNO) (Li et al., 2021). The Transformer model of interest is a drop-in replacement of the baseline U-Net, and it is named by U-Integral Transformer (UIT). UIT uses the kernel integral inspired attention (9), and we also compare UIT with the linear attention-based Hybrid U-Transformer in Gao et al. (2021), as well as a Hadamard product-based cross-attention U-Transformer in Wang et al. (2022). An ablation study is also performed by replacing the convolution layers in the U-Net with attention (9) on various mesh grid sizes, e.g., U-Net with an attention block added in the coarsest level. For more details of the data setup, training, evaluation in the all experiments please refer to Appendix B.
The comparison result can be found in Table 1. Due to the fact that FNO keeping only the modes in the lower end of the spectra, it performs relatively poor in this EIT benchmark that needs to recover traits that consist of higher modes (sharp boundary edges of inclusion) from lower modes (smooth harmonic extension). Thanks to Theorem 2, attention-based models are capable to recover “high frequency target from low frequency data”, and in general outperform the CNN-based U-Net despite having only of the parameters. Another highlight is that, thanks to the unique PDE-based feature map through harmonic extension, the proposed models are extremely robust to noise. The proposed models can recover the buried domain under a moderately large noise (5%) and an extreme amount of noise (20%) which can be disastrous for many classical methods.
|Relative error||Position-wise cross entropy||Dice coefficient||# params|
For a boundary value inverse problem, we propose a novel operator learner based on the mathematical structure of the inverse operator and Transformer. The proposed architecture consists of two components: the first one is a harmonic extension of boundary data (a PDE-based feature map), and the second one is a modified attention mechanism derived from the classical DSM by introducing some learnable non-local integral kernels. The evaluation accuracy on the benchmark problems surpasses the current widely-used CNN-based U-Net and the best operator learner FNO. This research strengthens the insights that the attention is an adaptable neural architecture that can incorporate a priori mathematical knowledge to design more physics-compatible DNN architectures. However, we acknowledge some limitations: similar to other operator learners, the data manifold on which the operator is learned is assumed to exhibit certain low-dimensional attributes that can be reasonably well approximated by a fixed number of bases.
This paper is reproducible. Experimental details about all empirical results described in this paper are provided in Appendix B
. Additionally, we provide the PyTorch(Paszke et al., 2019) code for reproducing in the supplemental material. The dataset used in this paper is available at https://www.kaggle.com/datasets/scaomath/eit-transformer. Formal proofs under a rigorous setting of all our theoretical results are provided in Appendices C-D.
This work is supported in part by National Science Foundation grants DMS-2012465 and DMS-2136075. No additional revenues are related to this work.
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Appendix A Table of Common-used Notations
|,||admissible Sobolev spaces for the inverse problems, which are|
|and for EIT with data pairs|
|the gradient vector of a function,|
|the delta function such that , .|
|normal derivative of , measures the rate of change along the direction of|
|NtD mapping from Neumann data|
|(how fast the solution changes toward the outward normal direction) to|
|Dirichlet data (the solution’s value along the tangential direction)|
|an underlying spacial domain in|
|a subdomain in (not necessarily topologically-connected)|
|,||’s and ’s boundary, -dimensional manifolds|
|,||the Sobolev space of functions|
|,||the bounded linear functional defined on|
|all such that ’s integral on vanishes|
|the seminorm defined for functions in|
Appendix B Experiment Set-up
b.1 Data generation and training
In the numerical examples, the data generation mainly follows what were featured in (Fan & Ying, 2020; Guo & Jiang, 2020). For examples please refer to Figure 2. The computational domain is set to be , and the two media with the different conductivities are with (inclusion) and (background). The inclusions are generated by four random ellipses with the length of the semi-major axis and semi-minor axis sampled from and , respectively. The rotation angle sampled from . There are 10800 samples in the training set, of which 20% are reserved as validation, and 2000 in the testing set for evaluation.
The noise below (13) is assumed to be
where specifies the percentage of noise, and
is a normal Gaussian distribution independent with respect to. As is merely pointwise imposed, the boundary data can be highly rough even though the true data is smooth.
Thanks to the position-wise binary nature of
, another choice of the loss function during training can be the binary cross entropy, applied for a function in , to measure the distance between the ground truth and the network’s prediction
Thanks for the Pinsker inequality (e.g., see (Cover, 1999, Section 11.6)), serves as a good upper bound for the square of the total variation, which can be further bounded below by the -error given the boundedness of the position-wise value.
The training uses 1cycle (Smith & Topin, 2019)
learning rate strategy with a warm-up phase. A mini-batch ADAM iterations are run for a total number of 50 epochs with no extra regularization such as weight decay. The learning rate starts and ends with, and reaches the maximum of at the end of the -th epoch. The . The result demonstrated is obtained from fixing the random number generator seed. Some testing results can be seen in Figure 5
. All models are trained on an RTX 3090 or an A4000. The codes to replicate the experiments are open-source and publicly available111https://github.com/scaomath/eit-transformer. .
b.2 Network architecture
At each resolution, The 2D Euclidean coordinates of a regular grid are applied a channel expansion through a learnable linear layer, and then are added to each latent representation.
Double convolution block.
The double convolution block is modified from that commonly seen in Computer Vision (CV) models, such as ResNet (He et al., 2016). We modify this block such that upon being used in an attention block, the batch normalization (Ioffe & Szegedy, 2015) shall be replaced by the layer normalization (Ba et al., 2016) which can be understand as a learnable approximation to the Gram matrices’ inverse by a diagonal matrix.
Coarse-fine interaction in up blocks.
The convolution layer on the coarsest level is replaced by the attention with pre-inner-product normalization in Section 3.2. The skip connection from the encoder latent representations to the ones in the decoder are generated using an architecture similar to the cross attention used in (Petit et al., 2021). and are generated from the latent representation functions on the same coarser grids, such that the attention kernel to measure the interaction between different channels is built from on the coarse grid. is associated with a finer grid. The major differences of ours with the one in Petit et al. (2021) are that ours are inspired by the kernel integral for a PDE problem, thus the modified attention in our method has (1) no softmax normalization or (2) no Hadamard product-type skip connection.
Appendix C Proof of Theorem 1
Theorem 1 (A finite dimensional approximation of the index map).
Suppose the boundary data is the eigenfunction of corresponding to the -th eigenvalue , and let be the data functions generated by harmonic extensions
where . Define the space on (the spatial dimension ):
Then, for any , there exists a sufficiently large such that
By Theorem 4.1 in (Guo & Jiang, 2020), we can show that
As , it is increasing with respect to . Then, there is a constant such that , . Given any , there is an interger such that , . Define
Note the fundamental inequality , . Then, if , there holds
if , there holds