A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations
Authors: Naveen Gupta
Medha Sawhney
Arka Daw
Youzuo Lin
Anuj Karpatne
Venue: ICLR 2025 Poster (2025)
Keywords: Machine Learning
Inverse Problems
Full-Waveform Inversion
Seismic Imaging
ML4Science
OpenFWI
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Latent space Models for jointly solving forward and inverse problems for subsurface imaging

Abstract: In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest to leverage recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets. The code is available at https://github.com/KGML-lab/Generalized-Forward-Inverse-Framework-for-DL4SI

Published as a conference paper at ICLR 2025
A UNIFIED FRAMEWORK, FOR FORWARD AND INVERSE PROBLEMS IN SUBSURFACE IMAGING USING LATENT SPACE TRANSLATIONS
Naveen Gupta't, Medha Sawhney' Arka Daw Youzuo Lin", Anuj Karpatne* Virginia Tech 2Oak Ridge National Lab UNC at Chapel Hill
ABSTRACT
In subsurface imaging, learning the mapping from velocity maps to seismic wave- forms (forward problem) and waveforms velocity (inverse problem) is impor- tant for several applications. While traditional techmiques for solving forward and inverse problems are computationally prohibitive, there is growing inter- est in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the va- riety of architectures explored in previous works, several open questions remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward- Borward-Inverse (GFT) framework, building on the assumption of manifolds and latent space trans- lations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain trans- lation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of- the-art performance for forward and inverse problems on a wide range of syn- thetic datasets and also investigate their zero-shot effectiveness two real-world- like datasets. The code available at ht /ttp::::///gttbb../// com/ com/KGML-aab. Generalized- Forward wwwpwward-Ivveeeeee Inverse Framework for DL4SI
INTRODUCTION
The goal of subsurface imaging is to identify the structure and geophysical properties of layers underneath the Earth's surface such as velocity maps (v), which useful for a number of applications including energy exploration, carbon capture and sequestration, and developing earthquake early warning systems (Zhang et al. 2013). A typical approach for subsurface imaging is to conduct energy generated from
seismic surveys, where the elastic made to propagate through the Earth's 's layers that are refracted, reflected, and received as seismic waveforms (p) the surface by multiple receivers. Mathematically, the relation between velocity maps and seismic waveforms is governed by the following acoustic wave equation:
more controlled sources is
0² (2p(z,2,t) 0² x) de p(a, 2, s(z, 2,t),
where v, and É represent the horizontal direction, depth, and time, respectively, and 2 is the Laplacian operator. Using Equation [ the forward problem subsurface imaging to compute seismic waveforms p given velocity maps by learning the forward mapping fu-p - P. The inverse problem, also referred to as Full Waveform Inversion (FWI) is then to learn the reverse mapping fp-0 V for inferring velocity maps e given observations of p.
"Equal Contribution. Corresponding authors: dawa @ orni. gov, (naveengupta, medhaj @ vt.edu

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While solving the forward problem requires expensive numerical solutions (Tago et al. 2012) of Equation [] learning the inverse is even more computationally demanding as it involves iterating over velocity maps model and ground-truth observations of (aarattlla 1988). As a result, traditional techniques for solving forward and inverse problems in subsurface imaging are difficult to scale in operational settings at the desired resolutions. In response, there is a growing interest in the geophysics commu- nity leverage deep learning for subsurface imaging (DL4SI), in particular, using image-to-iman translation methods to map seismic waveforms to velocity maps and back directly from data.
minimize the difference between simulated waveforms from the forward
For example, several deep learning methods have recently been proposed for FWI using encoder-decoder architectures such as Inver- sionNet (Wu & Lin, 2019) and VelocityGAN (Zhang et al. 2019). where the encoder maps seismic data to latent feature representations that are decoded back to the velocity domain (Lin et al.) (023). Techniques for solving the forward problem include the seminal work on physics-informed neural networks (PINNs) (Raissi et al. 2019) and recent works such as Fourier Neural Operators (Li et al. 2020). A recent work has also pursued solving both for- ward and inverse problems together using common architecture, Auto-Linear (Feng et al.) 2024). where two auto-encoders are first separately trained to project velocity and waveform fields to higher-dimensional manifolds, which are then linked via linear transforms for manifold translation.
Velocity Maps
Forward Problem
Selsmic
Wavaforms
F
Ep
Lg
+
F
Dv
Dp
Inverse Problem
$0..0
Generalized Forward-Inverse Framework
Figure I: A unified framework for solving for- ward and inverse problems in subsurface imaging.
Despite the variety of architectures explored in previous works in DLASI, several open questions still remain unanswered such as: (1) How does the size of latent spaces in encoder-decoder frameworks influence the quality of domain translations in DL4SI? While previous works such as InversionNet used lower- -ivr--dimnsional embeddings in their bottleneck layers, recent methods such as Auto-Linear mappings. (2) Do we need complex architectures for latent space
advocate for translations are linear transforms sufficient? (3) What is the role of manifold learning in DL4SI where latent spaces are used not just for translation but also for reconstruction? (4) Can we simulta- neously solve both forward and inverse problems using joint frameworks?
To answer these questions, we propose a unified framework to systematically characterize prior research in DLASI termed the Generalized Vorward-Inverse (GFI) framework (see Figure (T). The GFI framework builds on two key philosophies that are at the core of prior research in DL4SI. First, we assume that both velocity and seismic waveforms p can be projected latent space manifolds 8 and P using encodrrddecodr pairs that allow reconstruction of the original domains, referred to as the manifold assumption. Second, we assume that it is possible to learn bidirectional translations between the two manifolds i and referred to the lafent space trunslation assumption. We show that GFT encompasses previous works in DLASI including InversionNet, VelocityGAN and Auto-Linear, which can be viewed as specific instantiations of GFI. This unifying perspective helps us analyze the field of DL4SI going beyond specific architecture choices and also discover novel formulations in DL4SI such those explored in this paper.
In particular, we propose two new architectures within the framework of GFI: Latent U-Net (see Fig- ure 2a and Invertible X-Net (see Figure 2b). Latent U-Net uses two separate U-Nets (Ronneberger et al. 2015) to model latent space translations in both directions. This takes inspiration from the success of U-Nets in mainstream computer vision applications explore their benefits in DLASII Invertible-XNe uses Invertible U-Net (TU-Net) (Etmann et al.) 2020) to simultaneously learn both forward and inverse translations in the latent space using a single architecture. Invertible U-Nets builds on the idea of Invertible Neural Networks (INNs) (Ardizzone et al.) 2018)) to learn bijective mappings between input and output domains in DL4SI. We show that both Latent U-Net and Invert- ible X-Net achieve state-of-the- art (SOTA) performance for forward and inverse problems on wide range of benchmark datasets commonly used in DL4SI. We show that jointly solving forward and inverse problems using Invertible X-Net especially helps in improving the accuracy on the forward problem. We also investigate the zero-shot effectiveness of our proposed approaches on synthetic and real- world-like datasets, contributing novel insights in the field of DL4SI.
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Velecity Haps
Forward Prottem
Selamic Waveforms
Velucity
Forward Problem
Selsii Warelfrrm
Ner
Het
PlS-
((su4)
P
U-Nul
Cone Abr
Come Net
Cont Net
Cort Nef
12
1281000070
120x/30070
Probtem
300.
Inverse
Problem
(a) Latent U-Net architecture
(b) Invertible X-Net architecture
Figure 2: Schematic representation of proposed Latent U-Net and Invertible X-Net model.
2 RELATED WORKS
2.1 DEEP LEARNING FOR SUBSURFACE IMAGING (DL4SI)
Solving inverse problems in DL4SI has received considerable attention in recent years, thanks to the seminal work by Deng et al. (2022) in creating OpenFWI, an open-access synthetic dataset involving paired examples of velocity and waveforms with varying levels of complexity. This has enabled the formulation of FWI as an image-to-image translation problem, where the relationship between waveforms and velocity is Icarned through supervised, unsupervised, self-supervised methods (Lin et al. 2023). One of the early works in supervised learning for FWI includes InversionNet (Wu & Lin 2211)), which uses encoder-decoder architecture to project waveforms into flattened vector representations acting as the bottleneck layer, which are decoded back to velocity maps. VelocityGAN (Zhang et al. 2019) builds on the idea of InversionNet by using adversarial training scheme for learning the model parameters along with prediction loss. These works have also been extended for estimating 3D velocity fields in Feng et al. (2021).
Since ground-truth observations of velocity maps are difficult to obtain in real-world settings, a number of unsupervised and self-supervised formulations have also been explored for FWI. This includes the framework of UPFWI (Jin et al., (202I) that uses the supervision of physics-based forward model of the wave equation to enforce sycle-consistency with data-driven inverse model. While the training of UPFWI is completely unsupervised and achieves comparable performance as InversionNet in some settings, requires computationally expensive simulations of forward model at every training epoch, restricting its scalability. Recent advancements, such as IFWI Du jet al. (2024), FWIPLR (Sun al. (2023), and FWIRLG Yang & Ma (2023), integrate physics-based constraints and deep learning tackle challenges like low-quality data and geological complexity. A recent line of work in semi-supervised learning for FWI involves the framework of Auto-Lincar (Feng al. 2024). In this framework, both velocity and waveforms are first mapped to higher- dimensional manifolds using two auto-encoders separately trained to minimize reconstruction loss in their respective domains. This is followed by supervised learning of translations between the manifolds using two independent linear transforms (one for forward and one for inverse), building on previous observations of linearity in high-dimensional spaces for FWI (Feng et al. 2022.
2.2 DOMATN TRANSLATION METHODS
The goal in domain translation (Tsola et al. 2017 Zhu al., 2017 Choi et al. 2018 is to learn the mapping between input and output domains (c.g.. the CT scan of patient and the organ segmenta- tion from the scan), either in a paired or unpaired setting. A recent work on Latent Space Mapping (LSM) MMyee et al. 2023) has introduced a unifying framework for domain translation based on the manifold assumption (i.e., both input and output domains reside in low-dimensional manifolds learned by minimizing reconstruction loss) and shared latent space assumption (i.e., both manifolds reside in shared latent space making it possible to translate from one manifold to another). We take inspiration from L.SSs in domain translation develop our proposed framework of GFI.
3 PROPOSED METHODOLOGIES
3.1 GENERALIZED FORWARD INVERSE FRAMEWORK
As shown schematically in Figure T our proposed framework of GFI unifies the learning of the forward problem fo-pp V P and the inverse problem fp-r P V using a common archi-
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tecture involving latent space translations. The GFI framework is inspired by two key assumptions. First, according to the manifold assumption, we assume that the velocity maps u E V and seismic waveforms p E P can be projected to their corresponding latent space representations, 7 and p, respectively, which can be mapped back to their reconstructions in the original space, i and ý. Note that the sizes of the latent spaces can be smaller larger than the original spaces. Further, the size of ū may not match with the size of P (c.g., in Auto-Linear). Second, according to the latent space translation assumption, we assume that the problem of learning forward and inverse mappings in the original spaces of velocity and waveforms can be reformulated as learning translations in their latent spaces. We provide theoretical justifications for the existence of these latent space translations in the Appendix A Note that these assumptions can also be relaxed in specific instantiations of GFI.
Formally, 1et E. V - V and Ep P - P be the encoders that map velocity and waveforms to their respective latent representations, ū = (v) and p Ep(p). Similarly, let D, V + V be the decoders that map the latent representations back to the original spaces. Additionally, let Li- -ー P and LØ--;: P - V be the latent space translations between ū and ÿ and vice versa. The forward and inverse mappings between velocity and waveform p can then be formulated as:
and D, P +
-= (-)) Dy Li- (v) 6 = fprr = D2 o T (p)
(2)
(3)
Learning GFI with Paired Data:
Let us assume that we have a paired i..d. dataset (v. p)
q(u, p). where q(v, p) is a joint probability distribution. We can then learn the parameters of the GFI framework a supervised manner by optimizing the following loss functions:
min 9, Bug sp Boy
Potv..)F(Dp LE-s (U), C(p
(4)
min B O Oligo Bo,
L(D, o LE- o (p) = L(t
(5)
where L denotes a loss function such as the Mean Squared Error (MSE) loss, Mean Absolute Error (MAE) loss, even an adversarial loss if trained with a discriminator.
3.2 PRIOR WORKS IN DL4SI AS SPECIAL CASES OF GFI
In this section, we revisit previous works in DLASI and describe them using GFI terminologies, i.e., modeling mode (forward/inverse), size of latent space, whether they use manifold learning or not, and the nature of latent space translations. Table summarizes prior works specifie realizations of the unifying GFI Framework. Schematic illustrations of all prior works in the framework of GFI are provided in the Appendix Figure 9
Table 1: Characterizing prior and proposed works using GFI Framework.
Model
Modeling Mode
Manifold Learning
Latent Spuce
Trraslation
Sizes of Latent Spaces
InversionNet
Inverse
Identity
Low
VelocityGAN
Inverse
No
Identity
Low
FNO
Farward
No
Fourier Layers
N/A
Forward and Inverse Disjoinlly
High and Different
Auto-Linear
Yes
Linear
Latent U-Net
Forward and Inverse Disjoinily
Possible
U-Net
Low and Same
(ours)
Invertible X Net (ours)
Forward and Inverse Simultancously
Implicit
TU-Net
Low and Same
InversionNet and VelocityGAN: Both InversionNet and VelocityGAN operate exclusively in the inverse mode, mapping seismic waveforms to velocity maps. They utilize the same encoder-decoder architecture, with the encoder Ep projecting seismic waveforms (p) to lower-dimensional latent space and the decoder D, reconstructing the velocity maps (v) from this representation. The sizes of the latent spaces are low-dimensional, determined by the bottleneck layer of the U-Net architecture.
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In terms of latent space translation, these models rely an identity mapping in the bottleneck layer, Lg-+ = I. However, we can also interpret them as having a U-Net for translation, if we view the first few and last few layers as the encoder-decoder pair while the intermediate layers perform translation. The sizes of the latent spaces are low-dimensional and they do not perform manifold learning since the latent spaces of j and i are solely used for translation and not for reconstruction.
Auto-Linear operates in both forward and inverse modes, using two disjoint latent space translations. It follows two-stage approach: first, encoder-decoder pairs Ey,, De, Ep and Dp) are independently trained using reconstruction loss to project e and into high-dimensional latent spaces (U and p), where dēm(O) * dim( Second, linear transforms Le- and L are trained for translation, keeping the encoders and decoders frozen. Auto-Linear leverages the manifold assumption as the latent manifolds and p are useful for reconstruction. The sizes of the latent spaces are higher-dimensionee designed specifically to facilitate linear mappings.
Auto-Linear:
3.3 PROPOSED MODEL: LATENT U-NET
We propose a novel architecture to solve forward and inverse problems using two latent space trans- lation models implemented using U-Nets, termed Latent U-Net. As shown in Figure 2a Latent
U-Net uses ConvNet backbones for both encoder-decoder pairs: (E,, D,) and (Ep, Dp). to project
and p to lower-dimensional representations. We also constrain the sizes of the latent spaces of T and p to be identical, i.c., dim ()) = dim(p). so that we can train two separate U-Net models to implement the latent space mappings Li-s and Lø- Note that InversionNet can also be viewed as a special case of Latent U-Net in inverse-only mode with subtle architectural differences such as absence of skip connections, design of encoder/decoder and translation layers, and different dimen- sions of ê and p.
We train Latent U-Net in an end-to-end manner to directly learn latent spaces useful for translation. This is in contrast to the two-stage training process employed in Auto-Linear where latent spaces are first pre-trained for reconstruction, followed by translation. Specifically, for the forward problem, the components Dp, Le-p, and Eg are trained together by optimizing Equation 4 while for the inverse problem, the components De. Lø and Ep are trained together by optimizing Equation 5 Since the training of forward and inverse modeling components of Latent U-Net are disjoint from each other, Latent U-Net does not explicitly learns manifolds useful for reconstruction. However, is possible to modify the training objective of Latent U-Net by adding reconstruction losses in both forward and inverse problems to perform manifold learning. See Appendix sections D.3. for details
3.4 PROPOSED MODEL: INVERTIBLE X-NET
We propose another novel architecture within the GFT framework termed Invertible X-Net to answer the question: "can we learn a single latent space translation model that can simultancously solve both forward and inverse problems?" Recently, Invertible Neural Networks (INNs) (Ardizzone et al. 2018) have been introduced as a specialized class of neural networks capable of learning bijective mappings between inputs and outputs. INNs are inherently invertible by design; once the INN is trained to learn forward mapping ʃ X + y' between inputs E X' and outputs y 3/ the inverse mapping f-:: + X' can be obtained for free. A variant of INN has also been developed for image-to-image translation problems using the U-Net backbone, termed IU-Net (Etmann et al. 2020). However, one of the key constraints of INN and IU-Net is that they require dim(X') = dim()), which is difficult to ensure in subsurface imaging problems (since velocity and waveforms reside in entirely different spaces).
This is where we can leverage the GFI framework to employ IU-Net in the latent spaces of veloc- ity and waveforms, which can be constrained to be of the same size (just like Latent- UNetss, i.e., dim(ū) = dim( By using a single IU-Net, we can simultaneously learn both forward and inverse translations in the latent space, I-p-0 and Li-sp- We refer to this formulation as Invertible "X"-Net (owing to the cross- cross-shape of translations between the latent spaces) defined as follows:
p = Lo))(() = fiu.Ne (0);
i = (p)r(0) = Siutae (p)
(6)
$ = Dp fIU.Ne (0) E. (v):
0 D, fiu (f) Epp
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The objective function for training Invertible X-Net involves prediction losses on both velocity maps and waveforms defined as follows:
Lx-Net = Lforward + Linverse
(8)
=
(. CDD, fIU-Ner (t), +
(hai))----.o (o [D,, o fiu-Na o (p),,
(9)
Note that Invertible X-Nets facilitate bidirectional training as both the forward and inverse mod- eling components are trained simultaneously. As a result, both encoder-decoder pairs of Invertible X-Nets implicitly learn manifolds useful for reconstruction. Also, Invertible X-Net offers several key advantages over baselines. First, it simultaneously addresses both the forward and inverse prob- lems within single model architecture, whereas other baselines typically require training separate models for each task (c.g., Latent U-Net and Auto-Linear), leading to greater parameter efficiency. Additionally, the use of IU-Net ensures that the mappings between the latent spaces of velocity maps and seismic waveforms are bijective, guaranteeing a one-to-one mapping between these represen- tations- - property not necessarily true for other models such as Latent U-nets and Auto-Linear. Further, the bidirectional training of the forward and inverse problems introduces strong regular- ization effect as the gradients of the forward loss Lroward affects the parameters of fIU-Net. thereby affecting both forward and inverse performance. Similarly, the gradients of the inverse loss Linverse also impact both forward and inverse performance by updating the parameters of fIu-Net- Further details about the Invertible X-Net architecture provided in Appendix Section D.3.2
Cycle Loss for Training Invertible X-Net: Another advantage of Invertible X-net is that the ar- chitecture be trained with unpaired examples using cycle-loss consistency (Zhu et al. (017)- We consider both variants of Invertible X-Net (with and without cycle loss) in our experiments to demonstrate its effect on generalization performance.
EXPERIMENTAL SETUP
4.1 DATASETS
We consider the OpenFWI collection of datasets (Deng et al.] (022). comprising multi-structural benchmark datasets for DL4SI grouped into: Vel, Fault, and Style Families. The Vel Family in- cludes four datasets with simple geological patterns, while the Fault Family contains four datasets with fault-like deformations, presenting greater modeling challenges. The Style Family uses style transfer methods to create complex geological settings. The waveform data is structured (# source X recording time x receiver length), resulting in shape of (5 x 1000 x 70) for five uniformly spaced contrast, the velocity maps are repre-
seismic sources and 70 receivers over 1000 milliseconds. sented as (1 x depth x receiver length), with a shape of (1 x 70 x 70) reflecting spatial dimensions of depth and horizontal coverage. We also conduct -iit-i--ddttitttionnn evaluations on the Marmousi (Martin et al. (2066) and Overthrust datasets (Aminzadeh et al. (996). They are standard bench- marks that closely mimic real-world scenarios and exceed the complexity of OpenFWI datasets.
4.2 MODEL ARCHITECTURES
Latent U-Net and Invertible X-Net: We use two variants for Latent U-Net: (1) (Small) with 1 17M parameters, and (2) (Large) with 33M parameters. For Invertible X-Net, we used an architecture with 24M parameters trained with and without cycle loss. See Appendix D.I and D.3 details. Baselines: For the inverse problem, we considered InversionNet, Velocity and Auto-Linear as baselines. We obtained their trained model checkpoints to reproduce their results. For the forward problem, we implemented FNO (Li et al., 2020) and a U-Net model (termed WaveformNet) as baseline models (see Appendix C for details), along with using Auto-Linear.
The input seismic waveforms were standardized using a StandardSealer to center the data with a mean and unit standard deviation, while the velocity data scaled through MinMax normalization. See Appendix D.2 for details on hyper-parameter settings. To make a fair comparison with baseline methods, we kept the original train-test splits and reported results in the unnormalized space of both velocity and waveforms. We used Mean Absolute Error (MAF), Mean Square Error (MSE), and Structured Similarity (SSIM) as evaluation metrics since neither metric alone is fully comprehensive. MAE captures pixel-level accuracy while SSIM highlights structural similarity.
Training Details:

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5 RESULTS AND DISCUSSIONS
Quantitative Comparisons: Figure 3a compares SSIM on velocity predictions while solving the inverse problem across different OpenFWI datasets. We can that proposed models, which capture complex non-linear relationships in the latent space using U-Net and IU-Net architectures, consistently outperform baseline methods InversionNet. VelocityGAN, & Auto-Linear across ma- jority of datasets. Overall, the large Latent U-Net model shows the best performance in inverse modeling followed by the small Latent U-Net model. On complex datasets such as Style family and CFB, Invertible X-Net with and without cycle loss are also comparable to the small Latent U- Net model. In addition to SSIM, we also evaluate baselines using other metrics reported in Table 7 Apppendix and compare MAE across models in Appendix figure 3a
Figure 3b compares the performance of baseline methods on the forward problem (see Appendix Table for other evaluation metries). We can see that both Latent U-Net (Large) and Invertible X-Net show better performance than most baselines across all datasets. However, in contrast to the trends observed in the inverse problem, observe that Invertible X-Net with and without cycle loss shows the best performance compared to Latent U-Net models. We further notice that for simple datasets such FVA and FFA, FNO has significantly higher SSIM compared other models. We later qualitatively explain this pattern by showing that FNO tends to capture dominant modes of the waveform distribution while neglecting subtler reflection patterns.
Invenionteet
Latent U-beet (Large)
Wevetormiiet
Labent U-bet (Large)
ValociyGAA
Invert X-Net Inowet tile Xobeet (Cycle)
FNO
Invert X-Net
Auttn-lineer
Auri- Inner
Inowst hie X-bet (Cycle)
Labonc U-tttt (Smali)
Labenc U-Nett (5malin
FVA
FVA
rv0
rvD
CVA
FFA
0.6
0.7
0.8
0.9
1.0
0.75
0.80
0.85
0.90
0.95
1.00
SSIM
SSIM
(a) Inverse problem
(b) Forward problem
Figure 3: Comparison of Latent U-Nets (Small and Large), Invertible X-Net, Invertible X-Net (Cy- cle) with different baseline methods across different OpenFWI datasets.
Qualitative Comparisons: Figure 4a we compare inverse model predictions on three datasets, CVB, CFB, and STA, choosing one from each family of OpenFWI. We observe that our proposed methods (Latent U-Net and Invertible X-Net) produce considerably better velocity maps than the two SOTA baselines, InversionNet and Auto-Linear. We chose these datasets they are complex and have high heterogeneity as the velocity rapidly changes in all directions. For heterogeneous velocity maps, the predictions of baseline methods smooth and they fail delincate sharp changes in velocities. On the other hand, Latent U-Net and Invertible X-Net are able to predict better velocity fields capturing rapid variations in both shallow and deeper parts (highlighted using black-boxes in

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the figure). Between Latent U-Net (Large) and Invertible X-Net, we that the performance of Latent U-Net is slightly better especially on CFB and STA datasets. Note that Invertible X-Net uses smaller number of parameters (24M) compared to Latent U-Net (Large) (33M). We provide additional visualizations of more baseline methods and across other datasets the Appendix
4b visualizations of model predictions for the forward problem three datasets. Since these datasets are heterogeneous, the recorded seismic waveforms exhibit convoluted interference patterns. In particular, can see that in all ground truth visualizations, there is a direct arrival in the shallower layers represented as a 'slanted line" signature. While capturing the direct arrival relatively simple, the primary challenge in solving forward problems is to capture the subtler reflections in the deeper layers, where even small magnitudes of errors can result in large differences in inferred velocity maps. We can see that while all baseline methods are able to predict shallow waveforms easily (like direct arrivals), methods such as FNO and Auto-Linear are struggling to predict the seismic waveforms, especially the deeper regions (highlighted using black-boxes the Figure). On the other hand, our proposed models consistently improve results over baselines across all datasets. Additional visualizations for the forward problem are provided in the Appendix H Building on these quantitative and qualitative results, provide key insights on four open questions in the field of DLASI in the following.
Invertille
PE Bamiieeee
TOT
OF STA UNF
PHE PITAOO
PSTTF INC
PD P0110
Pli 1:00 Ewart
Drad
Arti Bey
Gound Toch
We divio
FIE E
PAC CN
RAC 2013
(a) Inverse problem
(b) Forward problem
Figure 4: Visualization of model predictions for inverse and forward problems across different Open- FWI datasets: CVB, CFB, STA.
250
400
5tip
200
200
300
150
MAE
200
50
100
50
10
20
30
40
50 66
(4)
Dimension
(b)
Latent Dimension
Latent Dimension
Figure 5: Effect of latent dimensions and skip connections (across latent dimensions) on the per- formance of Latent U-Net (Large) (a and b) and Invertible X-Net (c). 'A' and 'B' show "low" and 'high' complexity scenarios for each data group.
5.1
KEY INSIGHTS ON OPEN QUESTIONS IN DL4SI
5.1.1 WHAT IS THE EFFECT OF LATENT SPACE SIZES ON TRANSLATION PERFORMANCE?
we investigate the impact of varying latent space sizes the quality of domain trans- lations using the Latent U-Net (Large) (a) and Invertible X-Net(c) models on the inverse problem. We compare MAE by reducing latent space dimensions from 70 x 70 8 x 8. We can see that the quality of translations is not affected much for 'A' -amily of OpenFWI datasets, while the 'B' family shows monotonically increasing trend in MAE as we reduce latent dimension. Since the 'A" -faamilyy
In Figure 5
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was intentionally created to represent simpler velocity distributions, the smallest latent space (8 x 8) is sufficient to estimate velocity maps. However, the 'B' -family of datasets represent more complex velocity distributions, requiring sufficiently larger latent space for the translation. This indicates that the ideal size of the latent space should be decided based on the requirements of the problem being solved.
5.1.2 Do WE NEED COMPLEX ARCHITECTURES FOR TRANSLATIONS?
We further study the role of model size on translation performance by comparing Latent U-Net (Small) and Latent U-Net (Large) with varying latent dimensions Figure T8b (b). We find that for given latent dimension, Latent U-Net (Large) outperforms Latent U-Net (Small), motivating architectures with sufficient complexity for domain translations. We further study the impact of using skip connections in the latent space translation models varying latent space dimensions in (h).. We can see that adding skip connections shows marginal improvements in performance for larger latent dimensions. However, when the number of dimensions is low (less than 20), can see a sharp degradation in performance by removing skip-connections. This indicates that smaller latent representations are weaker and thus insufficient for domain translation in U-Net frameworks without being augmented with skip connections.
5.1.3 WHAT THE ROLE OF MANIFOLD LEARNING IN DL4SI?
Manifold learning for latent spaces in domain translation follow three strategies: (1) the "Reconstruct then Translate" (Auto-Linear), where encoder-decoder pairs are first trained for reconstruction, followed by translation on frozen latent spaces; (2)eDirectly Translate" strategy (Latent U-Net), which jointly trains encoder- decoder pairs and translation models optimize latent spaces for translation; and (3) a hybrid "Reconstruct and Translate" approach, where the model is trained with both reconstruction and translation losses. To investigate their impact, we evaluate these approaches across varying training data fractions for OpenFWI families (Figure (). Our findings reveal that the "Directly Translate" approach consistently outperforms the other two, highlighting that latent spaces optimized solely for translation are most effective. Hence, mani- fold learning, while beneficial for reconstruction, may lead to suboptimal performance when translation is the primary goal. We see similar trend for Latent U-Net (Small) and Invertible X-Net shown in Appendix F.3 Figure 22
Recorstrucl then Translate
-4-
FM
CFA
Directly Transiate
STA
Reconstructe Transiate
FFA
MAE
10'
10a
10%
100%
Training Fraction
Figure Comparison of Latent U-Net (Large) across three training objectives: direct transla- tion, reconstruction followed by translation, and combined learning of both, evaluated at differ- ent fractions of training data.
5.1.4 IS IT USEFUL TO JOINTLY SOLVE FORWARD AND INVERSE PROBLEMS?
As discussed the quantitative comparisons of Latent U-Net (Large) and Invertible X-Net on the forward problem in Figure 3bb we surprising trend that Invertible X-Net generally outperforms Latent U-Net (Large) despite having smaller number of parameters. Analysis of its training behav- ior reveals that Invertible X-Net initially focuses solving the inverse problem, gradually shifting to the forward problem in later epochs. (see Appendix E..3 Figures 12 and [3] for visualizations). This is facilitated by the combined loss function, where gradients from the inverse problem enhance forward problem optimization once good velocity solution is achieved. This bidirectional learning enables Invertible X-Net to outperform Latent U-Net (Large). Further comparisons Appendix F.3 confirm that Invertible X-Net trained only on the forward problem performs worse than both Latent U-Net (Large) and the jointly trained Invertible X-Net, emphasizing the advantage joint optimiza- tion. With average ranks of 2.7 (MAE) and 1.9 (SSIM) for inverse problems, and 1.4 (MAE) and 1.1 (SSIM) for forward problems (Appendix Tables 7 and 9). Invertible X-Net offers a balanced and efficient solution for joint tasks with just 26M parameters. In contrast, Latent U-Net, while slightly better for inversc-only tasks, requires 70M parameters for both forward and inverse problems (35M x2), making Invertible X-Net the more practical and effective choice for the combined optimization.

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5.2 ZERO- SHOT GENERALIZATION RESULTS
Extrapolating across OpenEWI Datasets: To assess the out-of-distriutiio generalizability of bert-perfo models on unseen datasets, Figure we evaluate zero-shot performance of mod- els trained across all datasets (shown as rows) when tested across all datasets (shown as columns), for both forward and inverse problems. Here, the color intensity indicates difference SSIM perfor- mance of two competing methods. We can see that Invertible X-Net outperforms other baselines for both forward and inverse problems across majority of train-test cases. The only exception when the models are trained on FVB, relatively simpler dataset than other data families such as Vel and Vel. From a geophysics perspective, we expect to have better generalizability for models trained on complex geological settings and tested relatively simpler ones, as complex geological structures can be thought of as juxtapositions of simpler geological structures. Additionally, we compare the zero-shot generalization for all methods in detail in Appendix E.4 (Figures 14
Veloc ty_ SSM Difference
Veledit SSM Offference
Weveform 5SN Difference
Weveform SSIM Difference
Twening Dor
8 E.S.
Tranno Dalese
Evoliation Dataset
Eraluation Datasct
Evalustion Dataset
Bralustion Dataset
(a) Invertible X-Net InversionNet (Inverse)
(b) Invertible X-Net (c) Invertible X-Net vs
(d) Invertible X-Net vs
Auto-Linear (Inverse)
FNO (Forward)
Auto-Linear (Forward)
Figure 7: Zero-shot generalization comparison for both inverse and forward problem using SSIM
difference across all OpenFWI datasets. Red color indicates proposed method (Invertible X-Net) is better whereas blue color indicate otherwise.
MA 270000
(a) Inverse problem
(b) Forward problem
Figure 8: Zero-shot generalization (both forward and inverse) on Marmousi and Overthrust datasets using models trained on STA and STB datasets, respectively.
Extrapolating on Marmousi and Overthrust Datasets: We show the zero-shot generalizability of our models on the more complex and real- worldwordd geological settings of Marmousi and Overthrust in Figure Here, chose models trained the most complex OpenFWI datasets. i.e.. STA (for Marmousi) and STB (for overthrust) datasets, and compared their zero-shot performance in predicting velocity maps (inverse problem). We observe that none of the baselines are able to invert velocity maps at high resolution. However, the predictions of Latent U-Net and Invertible X-Net still able to delineate the sharp changes velocity maps especially in the shallow part of the data. On the forward problem, our models are able to accurately predict seismic waveforms for both Marmousi and Overthrust data relative baselines.
CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS
In this work, we introduced a unified framework for solving both forward and inverse problems in subsurface imaging, termed the
(GFI) framework. Within this frame- work, we proposed two novel architectures, Latent U-Net and Invertible X-Net, that leverage the power U-Nets and IU-Nets to perform latent space translations, respectively. Our study addresses several key questions left unanswered by prior research in DL4SI. Future research directions include training a single model for all datasets together, given that models trained on complex geological environments can effectively extrapolate unseen simpler geological structures.
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7 REPRODUCIBILITY STATEMENT
All the code required to train and evaluate the proposed methods, well as the baselines, has been uploaded to an anonymous GitHuu repository: htttps://githbb com//GML KGML- lab/ Seneralized-Fore zedizzzd-doaasa Forward Inverse Framework fontewokk-oo--eee for-foo------ DL4SI Eor-D ing processing code used in this work are sourced from the OpenFWI website: https:/ Spenfwi-lang gel.annggtituul gut.ub..... wo/docs/data. html The dataset is further described in Ap- pendix Section B and 2 The model architecture and hyper-parameter details for all proposed models and baselines thoroughly discussed in Appendix Section
The data and correspond-
8 ETHICS STATEMENT
We would like to emphasize that the primary goal of this paper is to facilitate scientific inquiry of DL4SI. While advancements in subsurface imaging have traditionally been used for energy explo- ration, there are several emerging applications of subsurface imaging with positive societal conse- quences such as Carbon Sequestration, Earthquake Hazard Warning, and Environmental Monitoring. Research in DL.4SI thus has similar characteristics as research in mainstream Al applications such as computer vision and language modeling, with a diverse range of societal, environmental, and economic impacts.
ACKNOWLEDGMENTS
This work was supported in part by NSF awards IIS- 2239328 and IIS 110-3323 We are grateful the Advanced Research Computing (ARC) Center at Virginia Tech for providing access GPU compute resources for this project. This manuscript has been authored by UT-BBttelle, I.C., un- der contract DE-ACU5-000R222755 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US gov- ermment retains a nonexclusive, paid-up. irrevocable, worldwide license to publish reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access these results of federally sponsored research accordance with the DOE Public Access Plan https://www.enerryy. wwwp///o--pbbiiccccceetoee
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APPENDICES
A THEORETICAL JUSTIFICATIONS FOR LATENT SPACE TRANSLATION ASSUMPTION
In this Section, we provide two Lemmas to theoretically support our Latent Space Translation as- sumption for the Generalized-Forvaa Inverse (GFI) Framework.
Lemma A.1 (Forward Latent Space Translation Assumption). Let f V + P be arbitrary forward operator mapping velocity maps V seismic waveforms P. Let Ey :V= and Dy V V denote the encoder and decoder for the velocity space V, respectively. Similarly, let Ep P P and Dp
P denote the encoder and decoder for the seismic waveform space P. Here, ô E v and ğ E P represent the latent space encodings. If we assume that the auto-encoder for the velocity optimal, i.e., Dy Ev(v) = à v, then there exists fanctional mapping in the latent space Le-p1 P.
Proof. Given the forward operator f V - P. by definition, for any v E V. there exists p E such that p [(v).
Let the latent space representations ū E V and E P be defined by the auto-encoders follows:
i (v(v). p == Ep(p), p (p)
(10)
(11)
To construct the latent space mapping L (T), consider the sequence of compositions involv- ing the encoders, decoders, and the forward operator f:
- = Ep(p)
Ep(((u) p f(u))
= =E((f(0) (assuming reconstruction: 0 = v)
=D()(D)(O)) (since è (o)
(12)
Thus, by definition of the composition of functions, the latent space mapping can be expressed as:
Le-p- = E fo
(13)
Lemma A.2 (Inverse Latent Space Translation Assumption). Let f-1 : P - V be an arbitrary inverse operator mapping seismic waveforms P to velocity maps V that unique. Let Eg P P and Dp P P denote the encoder and decoder for the seismic waveform space P, respectively. Similarly, let E., V. Here, p E P and V represent the latent space encodings. If we assume that the outo-encoder for the seismic waveform space is optimal, i.e., Dp Ep(p) R2 p. then there exists functional mapping in the latent space Lg -
V and D. : - denote the encoder and decoder for the velocity space
Proof. Given the inverse operator f-1 P + V. by definition, for any p E P, there exists v E V such that (p).
Let the latent space representations p e p. and 1 be defined by the auto-encoders follows:
p = (p),,
p = (p)
(14)
6 (u),
(
(15)
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To construct the latent space mapping Lp) consider the sequence of compositions involv- ing the encoders, decoders, and the inverse operator /-:
ū = Eo(v)
(p( (since v = f
(D) (assuming reconstruction: p)
= (D,())) (since Dy(
(16)
Thus, by definition of the composition of functions, the latent space mapping can be expressed as:
LØ Ey o of Dr
(17)
Dataset
Examples
Velocity shape
Waveform shape
FlatVel-A
30,000
(1, 70, 70)
(5, 1000, 70)
-eat
30,000
(1, 70)
1000, 70)
Curve Vel-A
30,000
(1, 70)
1000, 70)
Vel-B
30,000
(1, (0, 70)
(5, 1000, 70)
FlatFault-A
60,000
(1, 70)
(5, 1000, 70)
FlatFault-B
60,000
(1, 70)
(5, 1000, 70)
CurveFault-A
60,000
71,70,
(5, 1000, 70)
CurveFault-B 60,000
(1,
(5, 1000, 70)
Style-A
67,000
(1, 70)
(5, 1000, 70)
Style-B
67,000
(1, 70)
(5, 1000, 70)
Table 2: Statistics the number of samples, the size of the velocity and waveforms for each dataset in OpenFWI Deng et al. (2022).
B DATASET DESCRIPTION
The OpenFWI comprises multi-structure benchmark datasets of significant size that can be used for
solving full waveform inversion (FWI) using machine learning techniques (Deng al.) 2022). In particular, the repository contains major groups of data: (1) Vel Family, (2) Fault Family, and (3) Style Family. These groups represent simple to complex sub-surface geological settings with seismic velocity and waveforms information. The Vel family is the simplest geological patterns including four datasets (1) FlatVel-A (FVA), (2) Vel-- (FVB), (3) Vulve (CVA), and (4) CurveVel. (CVB). The difference between Fllt and Curve Vel is that the former represents low-energy geological environments where the rock layers are deposited horizontally and the latter consists of curved layers which are formed due structural deformation of flat layers. The Fault family also has four datasets (1) FlatFault-A (FFA), (2) FlatFault-B (FFB), (3) CurveFault-A (CFA), and (4) CurveFault-B (CFB), Unlike Vel datasets, the Fault familly contains fault-like deformations, which is fracturing of rocks under certain pressure conditions. Due to the presence of faults, the Fault family becomes more complicated and challenging model. The Style family has datasets (1) Style- A (STA), and (2) Style-B (STB). This dataset is generated using the style transfer method where the COCO dataset (Lin al. 2014) is set as the content images and the Marmousi dataset set as the style image. This is the most complex OpenFWI dataset it represents highly complex geological settings where the velocity is changing rapidly and abruptly. In summary, the details about the Vel and Fault datasets are described in table
The waveform data is represented as (# source x recording time x receiver length) whereas the velocity follow (1 x depth x receiver length) shape. In seismic surveys, the wave arrival time is recorded at the surface and thus, the waveform data for a single source represented as a function of time and receiver length. In total, there are 5 seismic sources uniformly spaced along the surface, and
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wavefields recorded by 70 receivers (uniformly spaced) along the surface for 1000 milliseconds. Therefore, the seismic wavefields are of the shape (5 x 1000 70). On the other hand, velocity maps represented as functions of spatial dimensions, depth and horizontal coverage, and thus have the shape (1 x 70 x 70).
C PRIOR WORKS IN DL4SI AS SPECIAL CASES OF GFI
Figure 9 provides additional schematic illustrations of prior works DL4SI such as InversionNet, WaveformNet, and AutoLinear special cases of our proposed GFI framework.
Fornerrd Prostam
Waveforms
Forward Prustam
Waweforms
Cerr
Core
Cent
pland
vixuur
Manilly
Abe
J0T
Ieverse Problam
Protumm
InversionNat
WaveformNet
Velecity
Forward Problem
Setsmic
Cose Rot
Linsar
Coverse
Probsem
(SL.)
AutoLinear
Figure 9: Prior Works exprerssed as GFI
D ADDITIONAL EXPERIMENTAL DETAILS
D.I CYCLE LOSS FOR INVERTIBLE X-NET
Given the velocity maps and the waveforms p, the predictions for the waveform and velocity can be obtained using the Invertible X-Net as follows:
fo-p( (v) Dp TE- (v) = fp- = D, LØ (p)
(18)
(19)
Now, the Invertible X-Net architecture can be further applied on á and create the following transformations:
= fu-p (0) = Dp Le-i (i) fp (p) D. Li-+ (p)
(20)
(21)
The cycle-loss for Invertible X-Net can be mathemattcally defined as follows:
Loyele C(p, P) + C(v, 0)
(22)
where L is the loss function that can be MSE, MAE Elastic Loss. Note that the formulation of Cycle-Loss does not rely on paired examples, and can applied on un-paired data as well. The combined loss function including cycle-loss can be given
Cx-Ne (Cycle) Livrward + Linverse + Luyele
(23)
For training Invertible X-Net (Cycle) model, loss function shown in Equation 23
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D.2 ADDITIONAL TRAINING DETAILS
For training, we normalize the velocity using min-max normalization and seismic waveform using standard normalization to rescale the data to mean 0 and standard deviation 1. Table S shows other hyperparameter details for training Latent U-Net and Invertible X-Net models on OpenFWI datasets.
Since the baseline models have different normalization schemes than our models, we compare model predictions during evaluation by unnormalizing the predictions original domains. For exam- ple, AutoLinear uses min-max normalization for velocity and a combination of log-normalization Fnom (loge (1 (x)) (r|)* *sign(a) followed by min-max in the log- nogmaazal domain for waveform. We unnormalized both the velocity and waveform predictions so that we can measure and visualize errors in the predictions in the original space, allowing easy comparison across models.
D.3 MODEL ARCHITECTURE
We summarize details related to model architecture, layers, and number of parameters related to seismic and velocity encoder-decoder architecture in Table [3] and regarding the latent models used for translation in Table 4 Further, we compare our model parameters with bascline models in Table
D.3.1
LATENT U-NET
Latent U-Net architectures are specifically adapted to subsurface imaging by employing U-Net- based encoder-decoder pairs designed to handle domain-specific input and output dimensions, such as velocity maps and seismic waveforms, while enabling latent dimension processing. Both the Large and Small variants utilize identical encoder and decoder designs, featuring 5 layers embedding dimension of 128x70x70,
en-
coder and 6 layers in the decoder. These layers operate with channel sizes ranging from to 128 in the encoder and 128 to (velocity) or 128 to 5 (wave- form) in the decoder.
The Large Latent U-Net incorporates depth levels and convolutional blocks per depth level in its latent space translation model, resulting in total parameter size of 34.96M. The Small variant, de- signed for reduced complexity, includes convolutional block per depth level in its latent translation model, reducing the total parameter size 18.1300 18..
D.3.2 INVERTIBLE X-NET
Invertible X-Net leverages an iUNet-based architecture to achieve bidirectional mappings, en- abling consistent forward weelocit-ttowavveee tions within shared framework. iUNet introduces invertible coupling blocks, ensuring bijectivity for the latent space translation model. Max pooling replaced with orthogonal convolutional fil- ters for downsampling, while upsampling is performed using orthogonal deconvolution filters to maintain invertibility.
and inverse (waveform-to-veleeie transforma-
While the latent space translation model in Invertible X-Net is fully invertible due to its iUNet structure, the encoder-decoder pairs are not invertible. Therefore, Invertible X-Net as whole not fully invertible. However, by employing two separate encoder-decoder pairs for velocity and waveform domains, Invertible X-Net achieves an architecture that enables bidirectional mappings between the two domains while preserving consistency.
Invertible X-Net features depth levels, with 4 invertible coupling blocks per level, and adopts an deccodr structure similar to Latent U-Net, with 5 encoder layers and 6 decoder layers, embedding dimension of 128x70x70, and channel sizes ranging from 8 to 128. The latent translation model iUNet consists of 25.78M parameters, contributing total model size of 26.06M.
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Table 3: Architecture Details of Seismic Waveform and Velocity Encoder-Decoder models.
#Embedding
Model
#layers
Channels
#Params
Dim
Velocity Encoder
5
128x70x70
[8, 16, 32, 64, 128]
11632
Velocity Decoder
6
128x70x70
[128, 64, 32, 16, 1, I]
27877
Seismic Waveform
5
128x70x70
[8, 16, 32, 64, 128]
55680
Encoder
Seismic Waveform
6
128x70x70
[128, 64, 32, 16, 5]
186497
Decoder
Table Architecture details of Latent U-Net and IU-Net latent space translation models
#Conv blocks/
Model
#Depths
#Params
Coupling Blocks
Latent U-Net IU-Net
34.68M
4
25.78M
Table 5: Hyperparameter details for training Latent U-Net and Invertible X-Net models.
Model
#Epochs
Optimizer
LR
LR Scheduler
Latent U-Net
450
Adam
2e-3
StepL.
Invertible X-Net
450
Adam
2e-3
StepLR
Table 6: Comparison of encoder, decoder, and latent model parameters for model (Latent U-Net and Invertible X-Net) with other baseline models. The parameters for Latent U-Nets and Invertible X-Nets calculated for Latent dimension 70.
#Vel Encoder
Vel Decoder
#Amp Encoder
#Amp Decoder
ITranslation
#Total
Model
Params
Params
Params
Params
Params
Params
FNO
7.38M
InversionNet
9.34M
35.76M
Identity
24.41M
VelocityGAN
9.34M
35.76M
Identity
24.41M
Autolinear
12.98M
9.98M
2.29M
10.18M
16.5K
35.45M
Latent U-mal(S
11.6K
27.88K
55.68K
186.5K
17.86M
18.13M
Latent U-Net[L Large)
11.6K
27.88K
55.68K
186.5K
34.68M
34.96M
Invertible X-Net
11.6K
27.88K
55.68K
186.5K
25.78M
26.06M
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InverslonNat VelccitGAA
Latent (Largo)
Wevefomweet
Lacent U-Net (Langes
Invertible X-Net
PNO
Invertible
(Cydee)
AUZo
Net (Cyde)
Laten: Net (Small)
Latent Net 15mall)
FVA
FVA
IV8
PV8
CVA
Detaser
FFA
Detoset
FFA
mTD
I78
STA
STA
10
10
10"2
10"1
Inverse problem
(b) Forward problem
Comparison of Latent U-Nets (Small and Large). Invertible X-Net, Invertible X-Net (Cycle) with different baseline methods across different OpenFW] datasets.
Figure 10:
E ADDITIONAL RESULTS
E.
ADDITIONAL INVERSE MODELING RESULTS
We provide more detailed comparison of our models with other baseline models in Table 7 Our proposed models consistently outperform baseline models on multiple datasets indicating superior generalizability on in-distributions examples.
Additionally, we also show zero shot generalization of models on the Marmousi and Overthrust dataset in Tables Our model Invertible X-Net shows generalizability in SSIM indicating that overall prediction has better geological understanding than other baseline models.
Table 7: Comparison of our models (Latent U-Net (Large) and Invertible X-Net) with other bascline models for the inverse problem across 10 OpenFWI datasets. The bold highlights the best perform- ing model that dataset.
Matric
Mixiel
FVA
FVB
CVA
CVB
FFA
FRR
CFA
CPR
STA
STR
Average Rank
InversinNeet
19.67
52.77
192.77
234.57
25,90
158.31
38,90
246.94
93.96
103.37
28
MAE
Auto Linear
12.16
7005
110.22
273.02
24.58
181.39
41.38
268.47
107.5
95.63
3.
Latant U-Na
9001
2811
48.8
123.56
30,91
92.91
27.05
185.22
67.00
71.41
1.3
Inverible X-Mri
23.80
34.94
57.86
130.81
90.45
94.11
198.56
181.03
67.09
121.42
23
ImvrsioNNet
1002.74
12271.62
36438,33
189044.46
4081.10
68214.92
9490.73
136327.4
23626.76
58622.35
28
MSE
Auso- Liner
1053.03
33900.57
42391.39
236457.48
5982.25
81789.80 80
| 13715/42
154713.12
31274.60
21819.14
3.5
Latast L-No
21692
6464.06
12689.6
91784.50
2344.22
38935.12
37711.8
93249192
14353.55
14564994
1.3
Inverible X Net
1245.52
6969.42
14639.66
96121.90
7719.53
32559.95
77105.41
86812.36
13106.96
31550.22
2.4
InvesteeNNal
0.9844
0.9461
| 0.8073
0/6736
0.9768
0.7208
09866
06136
0.8858
06614
3
SSIM
Auso-Lines
09887
03044
0.8056
0.6199
0.7
0.6665
09424
0.5695
0.422
0.7274
388
Lataat L-No
0.9977
0.5019
0.9273
0.8156
0.911
0.8515
0.9
0.6930
0,9298
0,9064
1.3
Invesible X Net
0.9917
0.9769
0.9135
0.8076
0.9826
0.8832
09316
0.7116
0.9300
0,7913
1.9
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Table 8: Comparison of our models (Latent U-Net (Large) and Invertible X-Net) on real world like datasets Marmousi, Overthrust, Marmousi Smooth, Overthrust Smooth for the Inverse Problem. The bold highlights the best performing model on that dataset.
Metric
Model
Marmousi
Overthrust
Marmousi Smooth
Overthrust Smooth
InversionNet
282.39
273.80
179.34
148.34
MAE 4
Auto-Linear
285.38
298.89
206.38
207.41
Latnet U-Net
322.13
264.40
242.71
198.53
Invertible X-Net
298.39
308.50
245.86
258.26
InversionNet
160084.5
135988.73
67978.40
37804.58
MSE 4
Auto-Linear
1.5911..41
157434.56
78207.07
71128.92
Latnet U-Net
217508.11 17
122398.64
112685.39
69122.58
Invertible X-Net
180250.47
179093.10
107344.46
124700.75
InversionNet
0.46
0.4519
0.6044
0.7217
SSIM f
Auto-Linear
0.438
0.4097
0.5423
0.6447
Latnet U-Net
0.438
0.4807
0.6371
0.7031
Invertible X-Net
0.504
0.4827
0.6633
0.6952
E.2 ADDITIONAL FORWARD MODELING RESULTS
Similar to the inverse problem, we provide detailed comparison of our models with other baseline models in Table T0 Our proposed models consistently outperform baseline models on multiple datasets indicating superior in-distributions generalizability.
Table 9: Comparison of our models (Latent U-Net (Large) and Invertible X-Net) with other baseline models for the forward problem across 10 OpenFWI datasets. The bold highlights the best perform- ing model on that dataset.
Metrie
Model
FVA
FVB
CVA
|
CVB
FFA
FF8
CFA
CFB
STA
STB
Average Rank
INO
0,0077
0.0385
0.0700
0.1405
0.0282
0.0829
0.0605
0.1075
0.0292
0.0379
2.4
MAE J
Auto-Lincar
0.0340
0.0906
0.0744
0.1537
0.0541
0.1048
0,0700
0.1356
0,0492
0.0584
3.9
Latent U-Net
0.0389
0,0473
0.0581
0,1098
0.0512
0.059
0.0576
0.0891
0,0269
0.0354
2.3
Invertible X-Net
0.0527
0.0532
0.0887
0.0457
0.0477
0.0404
0.0671
0.0235
0.0341
1.4
FNO
0.0004
0.0066
0.0259
0.0825
0.0089
0.0358
0.0251
0.0471
0.0041
0.0053
1.9
MSE 1
Auto-I inear
0.0084
0.0427
0.0303
0.1045
0.0217
0.0625
0.033
0.0866
0.0174
0.0173
3.7
Latent U-Net
0.0173
0.0127
0.0160
0.0489
0.0344
0.0175
0.0442
0.0297
0.0053
0.0082
2.6
Invertille X-Net
0.0022
0.0176
0.0175
0.0305
0.0185
0.0117
0.0126
0.0163
0,004
0.0087
1.8
FNO
0.9967
0.9781
0.8881
0.8354
0.9667
0.8702
0.9166
0.8160
0.9655
0.9417
3.1
SSIM
Auto-Lincar
0.9289
0.9038
0.8567
0.9470
0.9253
0.9569
0.9300
3.5
Latent Net
0.9764
0.9779
0.9237
| 0.89
0.9659
0.9283
0.9571
0.8636
| 0.9819
0.9702
2.3
Invertible X-Net
0.9887
0.9782
0.9559
0.9221
0.9744
0.954
0.9757
0.9156
0.989
0.9805
1.1
E.3 IMPORTANCE OF COMBINED LOss FUNCTION FOR SIERTIBLE-XNEE
In this section, we focus on the training of the Invertible X-Net model using combined loss func- tion (incorporating both the forward and inverse problems), as opposed to training it solely with a forward loss function. Figure II shows that when model is trained using only forward loss, then its performance falls short compared to the Latent U-Net (Large) model. This discrepancy can be attributed to the fact that the Latent U-Net has higher model complexity than Invertible X-Net, despite their similar sizes. Nonetheless, when the Invertible X-Net model is trained with combined loss (forward and inverse), the model is able to outperform Latent U-Net model with good mar-
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Table 10: Comparison of our models (Latent U-Net (Large) and Invertible X-Net) on real world like datasets Marmousi, Overthrust, Marmousi Smooth, Overthrust Smooth for the Forward Problem. The bold highlights the best performing model on that dataset.
Metric
Model
Marmousi
Overthrust
Marmousi Smooth
Overthrust Smooth
FNO
0.1484
0.2726
0.1077
0.1892
MAB 4
Auto-Linear
0.2818
0.3018
0.2821
0.2730
Latnet U-Net
0.1338
0.2502
0.1013
0.1875
Invertible X-Net
0.1425
0.2311
0.1056
0.2062
FNO
0.1110
0.580
0.0811
0.3495
MSE
Auto-Linear
0.4227
0.6203
0,5325
0.5593
Latnet U-Net
0.1116
0.5494
0.0927
0.4133
Invertible X-Net
0.1168
0.4026
0.08130
0.5049
FNO
0.8148
0.7404
0.9021
0.8447
SSIM 1
Auto-Linear
0.6344
0.672
0.6670
0.7192
Latnet U-Net
0.8343
0.7700
0.9143
0.8626
Invertible X-Net
0.827
0.7863
0.9112
0.8500
gin. This highlights the value of joint training, demonstrating that simultancously learning both the forward and inverse problems can lead to better results than learning the two models separately.
Figures 12 and we illustrate asymmetry in learning the translation for the forward and inverse problems using CVB and CFA datasets. From the figures, we observe that the model learns the inverse mapping in the initial epochs and gradually starts to learn the solution to the forward prob- lem in later epochs. Since the network optimizes the combined loss function on both velocity and waveform together, the gradients from the combined loss help the model to achieve better forward solution. This corroborates with hypothesis that the model trained combined loss is able to learn the connection between forward and inverse problem.
Latent U. Nei (Lame))
Insertible XNrl (Cycle) Invertible XNot (Forward Only)
Invertible XNes
6x 10-
MAE
CVA
PFA
CFA
STA
Dutasel
Figure 11: Comparison Latent U-Net (Large), Invertible X-Net, Invertible X-Net (Forward Only), and Invertible X-Net (Cycle) on the forward problem across various OpenFWI datasets.
E.4 ZERO-SHOT PERFORMANCE
In this part, we provide detailed insights into zero shot performance of our models (Latent U-Net (Large) and Invertible X-Net) with other baselines across all datasets. This investigation helps us understand overall oftttf-distribution generalization of a model and underscore the importance of learning the translation problem in the latent space.
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E.5 INVERSE PROBLEM
Figure [4] and 15 show generalization performance of Invertible X-Net and Latent U-Net (Large) models respectively on the inverse problem using MAE, MSE, and SSIM metries. As described in the main paper, we show models trained across all dataset as rows and evaluated across all test datasets as columns. For comparison, we evaluate metries such as MAE, MSE, and SSIM and calculate its difference between our models and other baseline models. For MAE and MSE, when the color intensity blue, our model show better generalizability and vice-versa whereas, for SSIM, when the color intensity is red indicates better generalization of our model and vice-versa.
From Figure [4| we observe that the Invertible X-Net shows superior generalization over the base- line models (AutoLinear, InversionNet, and VelocityGAN) across all metrics except on the FVB dataset. Figure [4| (d) compare Invertible X-Net with Latent U-Net (Large) model where we see that Invertible X-Net shows better generalization on complex datasets such CFB, STA, and STB datasets, while Latent U-Net is better on relatively simpler datasets such as CVA, CFA, and more. Similarly, Figure 15 shows the comparison of Latent U-Net (Large) model with other baseline mod- els. As expected, we observe that our model is able to generalize much better than all the baseline models.
E.6 FORWARD PROBLEM
Similar to the inverse problem, we analyze out-of-distribution generalization of our models against baseline models across all evaluation metrics.
Figures 16 and 17 compares the performance of our models Invertible X-Net and Latent U-Net (Large) against baselines. In Figure [6]] we observe strong generalization of Invertible X-Net over all baselines- AutoLinear, FNO, and WaveformNet. Figure [6 (d) shows the comparative perfor- mance of Invertible X-Net against Latent U-Net (Large) where we see that Invertible X-Net domi- nates overall across all metrics. In Figure [7] we compare the Latent U-Net (Large) model against same baselines as above. The figure indicates Latent U-Net (Large) model has much stronger gen- eralizability than baseline models consistently.
F ABLATION STUDIES
EFFECT OF VARYING LATENT SPACE SIZES
Figure 18a (a) shows how the performance of Latent U-Net model is affected with change in the latent space size. As the latent space size is reduced, the MAE and MSE metrics for mapping seismic waveform velocity is increasing, while the SSIM is decreasing. The impact is more pronounced on complex datasets such as B group of datasets. These datasets represent geologically complex datasets and therefore may require larger latent space to encode geological heterogeneity. In Figure 18b. we compare the large Latent U-Net with the small Latent U-Net model, while latent space size also reduced. We observe that the performance gap between the two models reduces as the latent space size is also reduced. This underscore the importance of latent space size for encoding the geological features for an effective translation.
Further, we provide visualizations of the first two primary PCA and I-SNE components of the ve- locity latent space in Figure 19 and 20 respectively. We take the encoder trained on a dataset and get the latent space encoding on several datasets. This visualization shows the in-distribution and out-of-distribution generality of our encoders and highlights the importance of the manifold for la- tent space translation. Overall, we observe that the larger latent space 70 x 70 have better structure than x 8, indicating that the ideal size of latent space should be decided based on the complexity of dataset and problem being solved.
F.2 EFFECT OF SKIP CONNECTIONS FOR LATENT U-NET
In this section, we study the impact of skip connections of Latent U-Net model for the inverse problem. Figure 21 shows how the MAE and MSE are increasing and SSIM is decreasing when the latent space size is decreasing from 70 x 70 to 8 x 8. We observe that the impact of skip connections more pronounced at smaller latent space size opposed to larger space.
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F.3 EFFECT OF MANIFOLD LEARNING
Figure 22 for Latent U-Net Small further elaborates on our finding that direct translation learning consistently outperforms the two-stage approach, where translation follows reconstruction.
G NOISE EXPERIMENTS
Table 11: Quantitative results on CurveFault-B with Gaussian noise of varying variance o² added during testing for the inverse problem.
Model
1e
o² te-5
o² - le-4
o' Te-4
PSNIL177 teetl
PSNRL7756741
MAE
MSE
SSIM
MAE
MSE
SSIM
MAE
MSE
5SDM
MAE
MSE
SSIM
MAE
MSE
SSIM
Lainet Nei
185.22
90248.64
0.0930
185.54
93538.78
0.6266
187.03
94674,546
0.6917
188.56
95777.85
0.6906
195.94
101464.7
0.6844
Deamalation
0.17%
0.31%
1.52%
Inveniile X Net
180.97
86466 25
071177
181.32
85673.38
0.1115
162.83
57674.14
0.7106
184.17
88585.86 86
0.7057
192.22
94728,94
0,7055
Desnudation (9)
6/
0.10%
0.24%
0.02%
1,02%
1.39%
0.15%
1.76%
2.45%
0.27%
6.21%
9.55%
1.3%%
LiterLineer
154713.00
0.5695
270.29
156201,11
0.5082
277.05
162156.0
1098M2.26
211016.29
0,319
Degnataiio (13)
6-)
0.67
0190/5
0.21%
4 81
1.109
6.789
9.41%
2.1858
21 0M66
16 77%
8 8.999
Table 12: Quantitative results on FlatFault-B with Gaussian noise of varying variance o² added during testing for the inverse problem.
Model
o' 0
- 1e
o² Se
o² = 1e-4
a" Se-3
PSNR.81.001
PSNR.77 15541
PSNE-2100
PSNNRAA7 DeVEl
MAB
MSB
SSIM
MAB
MSE
SSBM
MAB
MSE
SS0M
MAR
MSF
SSIM
MAE
MIE
SSIM
Lane: U.N
92.93
35923.46
0,8515
92.47
34137.49
0.8511
05.43
34907.52 52
0.8499
98.19
36002.58
0.5492
105.83
40770.50
0.8417
(egnalation
6-)
0.629
Inveriile Nei
94.05
12548.49
0.8532
96.23
33283.05
0.8529
104.75
36522.60
0.55
110,30
35016.04
0,5774
122.66
45500 85
0.8382
(Sg
1-
61
(-)
2339%
2,258
0.00%8
11.37%
12.209
0.37%
17.27%
198716
0.875
30,419
39,828
1.70%
Aste-innse
181.19
85937.79
Desnalation (%)
6-)
3.31%
4.32%
0.64%
13.68%
20.57%
2.98%
22.26%
34.97%
5.15%
56.25%
10%.
15.57%
Table 13: Quantitative results on CurveFault-B with Gaussian noise of varying variance 0² added during testing for the forward problem.
Model
Te-5
Te-4
PSNR-06602JB
PSNR-
PSNR
MAE
MSE
SSIM
MAE
MSE
SSIM
MAE
MSE
SSIM
MAE
MSE
SSIM
MAE
MSE
SSIM
Lamnet U Net
0.0891
0,0297
0.8636
0.0891
0.0297
0.8635
0.0892
0,0297
0.8634
0.0897
0,0301
0.8630
0.0966
0.0396
0.8565
Degrabition (%)
(-)
6-)
(-)
0%
0%
0.01%
0.13%
0. 1.900%
0.01%
0.60%
1.40%
0.06%
8.37%
33.47%
0.82%
Invertible X-Net
0U0671
0.0163
0.157
0U1672
0.0164
0.9156
0.0674
0.0164
0.152
0U0678
0.0166
0.1177
0.0719
0,0193
0.9000
Degradition (%)
(-)
63
(-)
0.07%
0.77%
0.009%
0.425
0.54%
0.045
1,00%
1,68%
0.10%
7.08%
18.18%
0.73%
Auto-Lincar
0.135
0.0866
08188
0.1356
0.0868
0.8187
0.1360
0.0573
0.8183
0.1365
0.088
0.8179
0.139
0:0919
0.8159
Degradation (%)
6-3
()
0.14%
0.73%%
0.05%
1.61%
2.53%
0.35%
H ADDITIONAL VISUALIZATIONS
Here, provide additional visualization of the waveform and velocity predictions for baselines and our models, namely Latent- (small). Latent-NNet (large), Invertible- XNee, and Invertible- XNet (cycle), for both forward and inverse problems. Please note that we show the prediction of seismic waveform and velocity in the original space by unnormalizing the predictions for every model (Figures 23
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Table 14: Quantitative results on FlatFault-B with Gaussian noise of varying variance r² added during testing for the forward problem.
PSNR=0600d8
PSNK-41
PSNR-16110U8
PSNR-3000388
MAE
MSE
SSIM
MAE
MSB
SSIM
MAE
MSE
SSIM
MAE
MSE
SSIM
MAE
MSE
SSIM
Lasent U Net
0.894
0,0175
0.9233
0.0894
0.0175
0.9283
0.0598
0.0179
0.9279
0,0506
0.0157
0.9271
0.0896
0,0325
0.9178
Degrabation (S3)
(-)
6-)
(-)
098
0%
0.68%
2.16%
0.04%
1.99%
6.99%
0.128
1687%
85.09%
1.12%
Invectible X-Ner
0.4777
0.017
0.540
0.1678
0.0177
0,9540
0.0452
0.0119
0,9536
0,4477
0.0122
0.9531
0.0535
0.066
0.9465
Degradation (%)
(-)
63
(-)
0.17%
0.33%
0.94%
L,88
0.04%
193%
4.26%
0.10%
12.16%
36.39%
0.80%
Auto Linear
0.1048
0,0625
0.8756
0.1051
0,0628
0.8734
0.1061
0.064]
0.8726
0.1072
0.0656
0,8717
0.112
0,0727
0.368
Degradation (%)
2.59%
2.24%
1 CODE APPENDIX
All the code required to train and evaluate the proposed methods, as well as the baselines, has been uploaded an anonymous GitHub repository: https://gitms: Generalized-Poxte Forward Inverse Framework for - DL4SI The data and correspond- ing processing code used in this work are sourced from the OpenFWI website: https:// Openfwi-land gighuut....... wo/dio//oosso
com/ Poom/GGML- lab/
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w
W
(a) Epoch:
W
(b) Epoch: 10
W
(c) Epoch: 20
Figure 12: Training of Invertible X-Net model the CVB dataset illustrating velocity and seismic waveform learning with epochs.
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(a) Epoch:
(b) Epoch: 10
(c) Epoch: 20
Figure 13: Training of Invertible X-Net model on the CFA dataset illustrating velocity and seismic waveform learning with epochs.
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NW no
24
C6 50
7V. 712 CA
06
I78
dA
5t
50
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05
TA
(c) Invertible X-Net WaveformNet
766
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8005
704
A003
3,00
3003
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Stx
-1.50
500
.18
518
575
(d) Invertible X-Net Latent-UNel (Large)
Figure 16: Out-of-distribttion zero shot generalizations for the forward problem of Invertible X-Net
with Antorinear, FNO, WaveformNet (U-Net like model), and Latent U-Net (Large).
29

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Published as a conference paper at ICLR 2025
76
76
2.90
-0.9
NV. nv
SKA
-06
FM.
Pit SMX
-05
(a) Latnet U-Net (Large) AutoLinear
Waveform SSN Offfrence
8:5
Wevefomn Difrrnnce
Wareform Diferance
8:4
6:2
6:0
2.90
-000
-0.00
-004
-006
-06
4008
(b) Latnet U-Net (Large) FNO
Weaafom DeMenence
4110
Vauaffrm MAE Oefermace
Whuafom Eifhewnce
405
CX
+62m
0%
2%
+0.50
na cn ms PF (6 dt
(NC
We 06 OF mM CN
(c) Latnet U-Net (Large) WaveformNet
Figure 17: Out-of-distribution zero shot generalizations for the forward problem of Latent U-Net with AntoLi FNO, and WaveformNet (U-Net like model).
30

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Published as a conference paper at ICLR 2025
Lalent U-N-l (Lange)
Lotent U Nes (Lerge)
Latent U-Set (Lang)
190
000
Dewhach
Latent Dimetision
Lesent Dimectaion
(a) Effect Latent dimension U-Net Large
Loteut U-Not (Small Large)
Taremt TI-Nes (Rmaal va Eargs))
Tareet V-Neet (Small w Largs)
344
10000
13900
Plo OkncBuudiBB
1000
S5TX
15000
5000
Staal
Lang
2500
30
21
Lasent
Dimetation
Latent Dimettion
(b) Effect Latent dimension U-Net Small comparison U-Net Small
Figure 18: Effect of the size of latent sizes the performance of large and small Latent U-Net (small and large) on the FWI datasets.
yairet
of Theson fre PRtrec EIEEAA CINMOO P000
Figure 19: Visualizing the PCA projection Velocity latent space for x and x 70.
31

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Published as a conference paper at ICLR 2025
Figure 20: Comparing the t-SNE plot of latent space for velocity encoder.
Latent Net (Large)
Latent U.Ner (Large)
Latent U-Net (Large)
000000
000000
FLAVEL
SSTM
120
30000
Ship No 530p
5000
Lateat: Daursaisu
10
51
Lotent Dimenoio
Lebeze
Latent U-Nes (Small)
Latent L-Nee (Shmall)
Latuni T-Not (Small)
3000
B00
No
30
30
Lasont Dimemsion
Lasont
Figure 21: Effect of the size of latent space and skip vs no skip connections on the performance of large and small Latent U-Net models across OpenFWT datasets.
32

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Published as a conference paper at ICLR 2025
Reconstract Translate
Reconstruct Translate
Directly Translate
CVA
STA
Reconstruct then Transiate
CVA
STA
Reconstruck Transiote
FEA
102
Directly Translate
FFA
MAE
MAE
10
100%
200%
Training Fraction
Training Fraction
(a) Latent U-Net Small
(b) Invetible X-Net
Figure 22: Comparison of Latent U-Net's and Invertible X-Net's performance across three learning objectives: translation directly, reconstruction followed by translation, and combined learning of both, evaluated at different training fractions.
33

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Published as a conference paper at ICLR 2025
Ground Truth
Inversionkee
Ve odIVSAN
Ao Uinear
Latont Simalh
Lasont Laaree)
Inverbll Xilett
Wwort die (iycle
Was 382 9917 Invers Inversionhet lonkee
Mas 246 9239
Hill: 280 2842 Au Linear
Mil: D01
NIlE: 5OC 1299
HAB: 290 1968
NAB: 272 7250
Ground nd Truth
We edit
Labert Wes iSmali
Labert Ui-Ne (Larpe)
Invert X-Nett
Vivertt XiMee Kcycle
SE 3E40 902
SOl 3000 376
SCIM 3AAXA
TZEN C-RE3
TZIN 0-333
Wermaus emozes
Ground
Inversiontes
(Smato
Invertble
3300
SS 0-223
SEM 0-223
5.7033
0.718S
SSAN 6.7023
SSIN: 0.8325
Orethnnt smooth
WOR 77377883
MAR
MAR 275-454 6046
MIR SC42
NAR 165
AIFE 720
Bround Truft
7ND
Wereesam Set
Auto-Lineer
Latlent LAKie Somall
Latart Labiet Cutrpe
Invertille A-iiet
Povertibee X-let (Cycle)
IT MAR: C.1338
MAT: 0.1888
MAT: 8.1277 Auto Limeer
MAD: 4320 LATNNG 35mall
MAG 1707
MAE 0.0039
MAI: 0.1023
Ground Truiin
HNO
Warefomm
LANAN
Invention X-Nut
Invartible X-Mut (Cycle)
1.
Doeitt Ar1
SURE TELRS
SAIN PUAR
SSAMI EIWAS
80
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TURE SHIVSS G.00E
DURT EXTE
DER 1409
9,020
0,0238
Wareform
Auto Liteer
LARIN
L-Nee (Larpell
Pooertible X-hett (Cycle)
MOt: 0.0022
Net: 0.353
NAB: 0.3065
MAD: CO55
MAC D436
Voc 9.0377
Mot: 0.0348
Figure 23: Zero shot generalization results of model trained on Style-A dataset on Marmousi and Overthrust dataset samples and their smoothened versions.
34

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Published as a conference paper at ICLR 2025
Ground Truth
Inversionhie
Ve odlyGAN
Ao Uiner
Latont Simahh
Labent Larre
Inverblle Kilet
Wwort die (iycle
Wis 382 4929
MAs: 112 8876
Mill: 296-0573
Mil 277-3184
Niki: 000 6853
HAN 200 S460
NAb:: 6:00 6116
Ground nd Truth
Invers Inversionhet
odily
tu Linear
Latert Wes iSmalo
Labert LLarge)
InvertBle X-lett
Vivertt XiMler (cydle
Sc 9038
S0a 3051
SERR A0V7 200
SCIM 2,433 PRC
TCEN CAST3
TZIN D-4713
Vermaus emozes
Ground
Inversionkee
odhvSAN
(imato
Invertble
3300
SS
SEI 6.-22
9.6375
SSEN 6.9034
Orethnnt smooth
3200
MOD 144 3155
MAR
MAR 201 4158
MIR 3592
NAR 5574
MAR 25R 7610
APF 05-08
Bround Truft
7ND
Wherelsmm Sel
AutoeLineer
Lattert LARIN Comall
Lataett Libiet Cutrpe
Invertille A-iiet
Povertilee A-let (Cyclel
0.1044
MAT: 6,2280 Wareffrr
MAR: 8.3335 Auto Limeer
MAD: 4343
MAG 1807
VAE 0.1289
MAI: 0.1937
Ground Truiin
INO
LATNN 35malll
Latann LAANN
Invention X-Nit
Invartible X-Mut (Cycle)
1.
SURE SEAR
SAIN SIRE
SUM E.STER
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Povertible (Cycle)
MOt: 6/253
NAT: 0.0500
NAB: 0,204
MAD: 01514
MAC 0800
VAc 9.323
MAt: 0.0355
Figure 24: Zero shot generalization results of model trained on Style-B dataset on Marmousi and Overthrust dataset samples and their smoothened versions.
35

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Published as a conference paper at ICLR 2025
93
-05
2O e SOGT
SZIN EOT5E
BC 0.812
SCIN 3-31E
SODI SEXD LLaggn)
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(Cycle)
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00
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-05
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MAEE BAR0S
MASI
IMME RAIIN
HAS IUSDRG
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Autto Uiner
Latenz U/ (5mi
Lateent UI (Large)
Invvrnt bie KrNet
Invembble 1Cycle)
-05
MAC 9 0.053
MAR: DENT
MAR ROMS
MAX: DARS
VAC BARRL
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2000
MAE Invortiontee
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2000
2400
2000
MAS 4.3433
MAC 23,8826
MAC 2-4387
MAG 36,3700
IAC 4.118
MAC 36-4013
NAC: 13.0240
Figure 25: Visualization predictions for forward and inverse problems on FVA dataset.
36

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Published as a conference paper at ICLR 2025
-05
SEIN SEIN-CO4TE
SSIN 3.3453
SODC 9TT0 LLMggo
SEIN 3-0788
10 S61S
MAGSS AND
EXX33
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MAR 18
Ground 2:th
FNO
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Latenz U-lver (5mm
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Invembile X:leet (Cycle)
-05
MAC 6/333
NAR: 0.0160
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NAL: 39 8403
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NAE TO 5530
MAC 77.7234
NAC: 5870
Figure 26: Visualization of predictions for forward and inverse problems on FVB dataset.
37

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Published as a conference paper at ICLR 2025
Ground
PNO
Vaseefemmett
Unear
iSmall
L-Neet
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Figure 27: Visualization of predictions for forward and inverse problems on CVA dataset.
38

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Published as a conference paper at ICLR 2025
Starpet
(Cyde)
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SCOR
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Figure 28: Visualization of predictions for forward and inverse problems CVB dataset.
39

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Published as a conference paper at ICLR 2025
Ground Truth
TRO
Auto U-llet Ismat)
Labert U-fhet
Iroeettille X-het
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MAE 270 3361 WelocbyG4M
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Ground Truth
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MIN 19-9819
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Figure 29: Visualization of predictions for forward and inverse problems on FFA dataset.
40

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Published as a conference paper at ICLR 2025
ESIR 3086
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Figure 30: Visualization predictions for forward and inverse problems on FFB dataset.
41

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Published as a conference paper at ICLR 2025
60
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Figure 31: Visualization predictions for forward and inverse problems on CFA dataset.
42

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Published as a conference paper at ICLR 2025
80
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MAc: 130-4157 MelocilyGAN
NAK: TUS 1I53
TAL 2011M90
MAG: LOND
MAD: JAUS Invert bie X-MIT
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MAS: 206-3251 IebociyCAA
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MAES E716
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NAE 180 8OG3
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NAE: 416O
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Figure 32: Visualization of predictions for forward and inverse problems on CFB dataset.
43

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Published as a conference paper at ICLR 2025
Ground Trug
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MAD 8 6787 Wavelormffrrt
HAT: 6.840
MAC 8
NAT: 0.3361
MAC 6003
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Inverticle Kchae (Oydae)
NAF 8.3974 TRO
MIR & 1011 Abwueformmirt
MAR BA167
MAFF R.5346
MAF MAT
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MAE 8 188L Wavelorm Best
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MIC 24.7546
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MIC 63 393999
MIG 39-8034
NAS 26-8870
NAE 27.7707
MAE 7500
MAB 48 0509
Figure 33: Visualization predictions for forward and inverse problems on STA dataset.
44

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Published as a conference paper at ICLR 2025
Viveltrmeltt
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(Cydl)
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PAC
MAS: 0.3055
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MAC: EC3D
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MAC NOVS
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SCNN 8-7013
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MAS 29 28.333
MAG $5.9550
VAE 03 3023
MiS RE0833
MOS 66-0019
MAG 41344566
MAG 6502
Figure 34: Visualization of predictions for forward and inverse problems on STB dataset.
45


Reviews:
[{"review_id": "anXE6PeCG3", "reviewer": "Reviewer_Lx2Y", "summary": "Subsurface imaging is a technique for identifying the geophysical properties of layers underneath the Earth's surface. There are two directions: learning the mapping from velocity maps to seismic waveforms, called the forward problem, and the inverse problem which is the mapping from seismic data to velocity maps. Most of the previous deep learning-based research in the field focuses only on the inverse problem also referred to as full waveform inversion (FWI). This paper proposes a generalized forward-inverse (GFI) framework that unifies both directions, i.e., FWI and the forward problem. The framework builds on the assumption of manifolds and latent space translations, proposing two model architectures for the latter, namely Latent U-Net and Invertible X-Net. Latent U-Net architecture employs two U-Nets for translation between the velocity and waveform latent representations and vice versa, while Invertible X-Net uses a single IU-Net that simultaneously learns forward and inverse translations. The GFI framework encompasses previous works in deep learning for subsurface imaging, at the same time trying to answer questions such as the effect of latent space size, the importance of manifold learning and the value of jointly solving forward and inverse problems. The models were evaluated on the synthetic OpenFWI dataset and their generalization ability was tested on two real-world-like datasets.", "questions": "1. Could you comment on the complexity of the problems and how to \"calculate\" it when choosing the latent space size? How to choose this size for real-world applications and are there any limitations in real-world scenarios?\n2. Why have you decided to remove Invertible X-Net from the experiments in Section 5.1.2.? Have you tried different model sizes for this architecture?  What is the relation between the number of parameters in encoders and decoders and the U-Net and how does it influence performance? \n3. It is not clear from the paper what are the benefits of having such a unified framework. Could you comment on the limitations? What would be your recommendation on which architecture to use and when?\n4. Could you comment on the examples in Figure 4. and Figure 8. and the performance of the models achieved in terms of MAE?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is clearly presented and well-organized.\n- Even though the GFI framework is not the first idea that unifies seismic imaging forward and inverse problems (e.g., Auto-Linear [1]), it tries to systematically characterize and unify prior research in this area. The paper answers a few questions that were open in the research field. The paper concludes the following: the size of the latent space should be decided based on the complexity of the problem, the training of the latent spaces should be influenced based on the translation objectives and jointly solving forward and inverse problems is helping the model to achieve better forward solutions. In my opinion, these are answers to important questions that were backed up by the experiments in the paper.\n- The paper introduces U-Net and IU-Net architectures for learning translations between velocity and seismic waveforms in the latent space. This leads to two new architectures for unified seismic imaging, namely Latent U-Net and Invertible X-Net. In comparison, the Auto-Linear framework's architecture comprises two separate linear layers trained for the forward and inverse translations in the latent space.\n- Extensive and systematic experiments. The proposed approach was compared to the existing state-of-the-art methods both for the forward and inverse seismic imaging problems. The proposed method employing Latent U-Net architecture outperforms the existing methods in most of the datasets in both forward and inverse directions.\n\n---\n[1] Y. Feng, Y. Chen, P. Jin, S. Feng, Y. Lin, \"Auto-Linear Phenomenon in Subsurface Imaging\n.\" ICML, PMLR:235, 2024, pp. 13153-13174.", "weaknesses": "- Section 5.1.1. \"What is the effect of latent space sizes on translation performance?\" is kind of vague. It states that the ideal size of the latent space should be decided based on the complexity of the problem being solved. It is not clear how to determine the complexity in real-world applications. I think the authors should be more specific in this section.\n- In Section 5.1.2. \"Do we need complex architectures for translations?\", there is a comparison between large and small Latent U-Net. The only difference was the complexity (size) of the U-Nets for the latent space translation and the influence of having skip connections. What is the relation between the number of parameters in encoders and decoders and the U-Net and how does it influence performance? The Invertible X-Net was not included in the experiments and only one size was used throughout the paper. It would be useful to see how its size influences the performance.\n- The Invertible X-Net model achieves worse results than the Latent U-Net model for the inversion problem and often fails to outperform other state-of-the-art methods. In contrast, it outperforms most of the methods for the forward problem. It is not clear from the paper what are the benefits of having such a unified framework. I think the limitations should be stated more clearly in the paper.\n- In the results section, in qualitative comparison, Figure 4. and Figure 8. are missing a measure of error (at least MAE). The picked images should have approximately the same MAE as the reported results in the tables presenting a realistic case and fair comparison. It is not clear whether the presented results were in a sense cherry-picked, e.g., the proposed method achieved better results than the average and other methods below the average for the illustrated example.\n- Instead of having a qualitative comparison of the methods on the two real-world-like datasets, I think it would be much more representative to have a quantitative comparison. Here, the presented examples might also be cherry-picked.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XkQm3KCQet", "reviewer": "Reviewer_VWh3", "summary": "This paper explores deep learning methods to address the forward and inverse problems in subsurface imaging by mapping between velocity maps and seismic waveforms. The authors introduce the Generalized Forward-Inverse (GFI) framework, which unifies past approaches and addresses open questions around latent spaces, manifold learning, and model complexity. They propose two novel architectures, Latent U-Net and Invertible X-Net, which achieve state-of-the-art results on synthetic datasets and show promising zero-shot performance on real-world-like datasets.", "questions": "please see my concerns in the Weakness", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper presents a unified architecture for addressing both forward and inverse subsurface imaging problems, while previous studies typically focus only on inversions. Modeling these problems together within a single framework could offer valuable insights, though it may also introduce potential challenges.", "weaknesses": "1. The paper’s narrow focus on subsurface imaging may limit its relevance for a broader audience at ICLR. If there are analogous problems in other fields where their framework could be applied? \n2. Comparisons to prior work are outdated, with main competing methods from 2019, like InversionNet and VelocityGAN. The authors are encouraged to consider more recent advances, such as “Physics-Informed Robust and Implicit Full Waveform Inversion Without Prior and Low-Frequency Information (2024)” and other state-of-the-art methods from 2023, for a fairer and more relevant assessment.\n3. While the authors aim to model the forward problem using neural networks, it is uncertain if the learned mapping can capture complex physical dynamics, as seen in the governing equations of seismic data. Extensive out-of-domain experiments are recommended to assess the model’s robustness and generalizability to real-world data.\n\n4. The paper's structure could be streamlined to highlight the proposed methodology. Currently, a substantial background section precedes a brief description of the methods. Specific sections of the introduction and related work and preliminary knowledge could be condensed.\n\n5. The model specifications lack detail; adaptations made to standard UNet for subsurface imaging and the mechanisms enabling X-Net’s invertibility are unclear. Clarifying these aspects and specifying when each model might be preferred would improve understanding.\n6. Experimental comparisons with recent baselines (published within the last 2-3 years) would add rigor to the evaluation, as several recent methods could serve as relevant benchmarks. To name a few, “Physics-Informed Robust and Implicit Full Waveform Inversion Without Prior and Low-Frequency Information 2024”; “Full-waveform inversion using a learned regularization 2023”; “Wasserstein distance-based full-waveform inversion with a regularizer powered by learned gradient 2023”; “Full-waveform inversion using a learned regularization 2023”, and more.\n7. Key numerical results in Section 5 are deferred to the appendix, which reduces clarity. Summarizing key findings directly in the main text would enhance readability.\n8. The Invertible X-Net significantly underperforms other models, raising questions about its ideal use cases. Further context on scenarios where the X-Net's design offers advantages would be valuable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3LLNM1JUSS", "reviewer": "Reviewer_74KD", "summary": "The paper proposes a new training scheme combined with two new model architectures for the problem of learning the forward and inverse map in deep learning for subsurface imaging. \nThe training scheme consists of jointly learning the encoders and decoders for the two domains (velocity maps and seismic waveforms) with the translation network transforming the latent space representations from one domain to another.\nThe two architectures differ in the characteristics of the translation network with one consisting of two separate unidirectional U-nets (latent U-net) and the other of a single bidirectional invertible network (invertible X-net).\nBoth architectures achieve a what seems to be significant improvement in performance over existing works.\nFinally the paper formulates a unifying framework in which existing approaches can be categorized and distinguished.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.\t(Quality) High quality of experimental evaluation: Experiments are extensive over a range of datasets including an extensive evaluation of out-of-distribution performance and qualitative results are provided in addition to quantitative results. Hence, the conclusion of generally improved performance seems sound. \n2.\t(Significance) The improvements in performance seem to be significant, however interpretability of how significant could be improved (see weaknesses point 5)\n3.\t(Significance) The finding that jointly solving forward and inverse problems is helpful for the forward problem but not for the inverse problem is interesting. \n4.\t(Originality) The proposed unifying framework helps to quickly understand the different existing approaches and will be helpful for future works to expand the framework or propose improvements within it.\n5.\t(Clarity) The paper is overall well written and easy to follow also for researches not actively involved in the field of subsurface imaging.", "weaknesses": "Major concerns\n\n1. (Clarity) The experiment 5.1.3 addresses the question ‘should we train latent spaces solely for reconstruction?’, which I assume corresponds to the question 3) formulated in line 083 ‘what is the role of manifold learning?’.  \n1.1 For increased clarity I’d suggest not changing the phrasing of the questions.  \n1.2 The setup is unclear to me. What are training fractions? A fraction of the training set or a fraction of the training time? And why is it instructive to look at 5% and 10% training fraction?  \n1.3 So, is the conclusion form the experiment that learning manifolds is not important? It is difficult to grasp the conclusion if the wording is not consistent. Is “influence the training of the latent spaces based on the translation objectives” identical to “not learning manifolds”?  \n1.4 To better answer the question I think the following baseline is missing that combines the two alternatives (reconstruct then translate and directly translate) into directly learning the translation networks while having at the same time reconstruction losses (basically what is described in lines 243-245). Seeing how this combination performs would help answering the question if manifold learning helps.  \n2. (Originality) The idea of the Latent U-net seems to be very similar to the InversionNet following the comparison in Table 1.  \n2.1 The Modelling Mode is the same because the forward and inverse part of the Latent U-net are completely disjoint and hence correspond to an InversionNet and a ForwardNet (as long as no manifold learning is performed).  \n2.2 While the latent U-net allows manifold learning, the experimental results pertain to the case without manifold learning same as the InversionNet.  \n2.3 Latent space  translation is said to be identity for the InversionNet and U-net for the Latent U-net, but couldn’t we also just claim some middle layers of the InversionNet to perform latent space translation?  \n2.4 The size of the latent space is low in both cases.  \n--> So to my understanding the difference between the two boils solely down to the architectural design and size of the network but not to any conceptual differences. If that is the case, it should be stated more directly in the paper and the exact architectural differences should be explained in more detail. \n3. (Clarity) From Section 5.1.4 it is not entirely clear what the answer to the question raised in the paragraph (Is it useful to jointly solve forward and inverse problem) is. Is the answer that no it is not useful for learning the inverse map but yes it is useful for learning the forward map (both answers based on the findings in Figure 3)?\n\nMinor concerns\n\n4. (Clarity) Please provide links to the exact Section in the Appendix rather than just referencing the entire appendix (e.g. lines 308, 311, 314, 377, 388). \n5. (Significance) Include quantitative scores in the qualitative examples (e.g. in Figures 4 and 8) to give a better understanding of how differences in scores translate to significantly perceptible differences.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Nst1SEJ4w9", "reviewer": "Reviewer_nmvk", "summary": "This work summarizes recent advances in deep learning-based mapping between velocity maps and seismic waveforms, introducing a unified framework (GFI) to systematically characterize prior research. Using GFI, key factors affecting mapping performance are investigated, alongside the proposal of two novel model architectures. Comprehensive experiments reveal the impact of these factors, with the proposed models achieving state-of-the-art (SOTA) performance in both forward and inverse problems.", "questions": "Invertible Neural Networks (INNs) often have limited flexibility in adjusting the dimensions of input and output, the number of layers, filters, and convolutional kernel sizes. These constraints can impede performance for complex tasks.\n\nWhen mapping between velocity maps and seismic waveforms—data from two distinct domains (one spatial and the other spatio-temporal), why Invertible X-Net could demonstrate superior performance?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This work is the first to investigate key factors influencing mapping performance.\n2. Extensive experiments were conducted to conclude the influence of these factors.\n3. Two novel networks are introduced for mapping between velocity maps and seismic waveforms, achieving SOTA performance.", "weaknesses": "1. This paper resembles a survey and might be more suitable for journal submission.\n2. A significant portion of the content is placed in the appendices, making it less reader-friendly.\n3. Aside from extensive experiments on factor influences, the work offers limited novelty and lacks theoretical analysis.\n4. Additional factors should be investigated, such as the robustness of data noise across different network architectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)