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Solving the Elastic Wave Equation with Physics-Informed Neural Networks: A Robust and Critical Assessment

Davide Staub, Ben Moseley

arXiv:2609.07983v1cs.LG

TL;DR

Repeated seismic simulations across sources in large, highly heterogeneous media are computationally demanding. The paper compares PINN architectures with different levels of wave-physics knowledge and conditions PINNs on source location, finding that encoder-decoder designs balancing physical knowledge and flexibility occupy the optimal architecture position.

  • Problem

    Repeated seismic simulations across sources in large, highly heterogeneous media are computationally demanding, motivating alternatives to traditional numerical methods.

    Source: §3.1 · p. 30

  • Method

    The paper compares PINN architectures with differing wave-physics constraints and conditions a network on source location.

    Source: §4.6 · p. 99, §5.1 · p. 101

  • Results

    Encoder-decoder PINNs balancing wave-physics knowledge and flexibility occupy the optimal architecture position; designs at either extreme show poorer convergence.

    Source: §4.6 · p. 99

  • Takeaways & Limitations

    Conditioned PINNs can process multiple seismic sources in one inference without retraining for each new source location.

    Source: §5.1 · p. 101, §5.7 · p. 123

    PINN-to-FD L2 comparisons have variance from discretization error in the FDM reference, which decreases as resolution increases.

    Source: §3.5.1 · p. 44

Abstract

from arXiv · show

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a meshfree alternative that integrates physical principles into the learning process. This presents a new paradigm compared to traditional discretization methods and purely data-driven machine learning techniques. While promising, PINNs are not a panacea; they inherit challenges such as spectral bias and unstable convergence. Moreover, their potential in seismology remains largely unexplored. In this work, we provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology. We investigate the performance of PINNs on problems with varying degrees of complexity across various seismic sources and parameter models, from constant to highly heterogeneous settings. A pivotal aspect of our work involves investigating whether embedding physical principles directly into the network architecture enhances convergence and accuracy. We test an extensive range of neural architecture designs, from unrestricted, uninformed PINNs to highly specialized ones. We find that integrating an understanding of wave physics into the network design significantly improves accuracy. For instance, introducing a custom wavelet or plane wave layer, coupled with encoder and decoder layers, consistently yields a relative $L_2$ error approximately half that of the standard PINN, as evidenced across numerous experiments. We further demonstrate that this novel architecture enhances accuracy when applied to the acoustic wave equation, underlying the versatility of our network. Another key contribution of our research is the successful conditioning of PINNs on seismic source locations. This signifies a considerable advancement towards rapid seismic hazard detection and seismic analysis.

Note on this arXiv version

The arXiv document substantially reproduces Davide Staub’s ETH Zürich master’s thesis, with the scientific content unchanged apart from updated front matter and minor typographical corrections.

Note on this arXiv version

  • The thesis was submitted on 5 March 2024; the arXiv author list is Davide Staub and Ben Moseley, and the thesis was supervised by Siddhartha Mishra and co-supervised by Ben Moseley.The thesis is available at doi:10.3929/ethz-b-000668359.Source: Note on this arXiv version · p. 2

Introduction

The introduction motivates evaluating PINNs for seismic wavefield simulation, where conventional methods can be computationally demanding and PINNs’ suitability for the elastic wave equation remains underexplored. It outlines an assessment across seismic settings and architectures, followed by chapters on baseline performance, architecture changes, and source-location conditioning.

1.1 Background and Motivation

  • PINNs are presented as meshfree surrogate models that may reduce simulation demands and infer seismic responses across space and time after training.The introduction contrasts this with mesh-based simulations that require a complete run for each source location.Source: §1.1 · p. 10
  • PINNs’ performance on the elastic wave equation, suitable architectures, and ability to scale to practical seismology problems remain insufficiently explored.The introduction identifies spectral bias as a challenge for resolving high-frequency components and multiscale problems.Source: §1.1 · p. 10
  • The thesis assesses PINNs for the elastic wave equation across parameter models and source conditions, including inference conditioned on source location.It plans to compare accuracy, efficiency, and scalability with conventional numerical methods and evaluate novel network architectures.Source: §1.1 · pp. 10–11

1.2 Thesis Structure Overview

  • The thesis proceeds from baseline PINN applications and limitations to architectural improvements and strategies for source-location conditioning.The overview names accuracy, hyperparameter-tuning intensity, and efficiency as baseline limitations.Source: §1.2 · p. 11

Background

The background situates elastic-wave modeling in seismology alongside conventional numerical methods and motivates PINNs as physics-constrained, meshfree PDE solvers for settings with scarce or noisy data. It outlines how PINNs are trained for forward problems, including their physical constraints, optional measurement data, and choices of architecture, differentiation, and optimization.

2.1.1 Introduction

  • Seismic waves reveal Earth’s interior structure, and for most applications they are approximated as elastic waves causing temporary, reversible deformations in Earth materials.The elastic wave equation governs their propagation and interactions with solid materials, including reflection, refraction, and diffraction.Source: §2.1.1 · pp. 12–13

2.1.2 Formulation

  • The study uses an elastic, isotropic displacement formulation with spatially varying Lamé parameters; external forces are represented through initial conditions, and viscoelastic relaxation variables are omitted.Under isotropy, the material parameters reduce to λ and µ, which determine pressure- and shear-wave velocities.Source: §2.1.2 · pp. 13–15

2.2.4 Numerical Baseline Used in This Thesis

  • Because no universal analytical solution exists for the elastic wave equation, PINN evaluations are benchmarked against numerical simulations; the study implements its displacement-based FDM solver using DEVITO.Unless otherwise specified, the implementation uses a 512×512 spatial grid and dt = 1e −3.Source: §2.2.1 · p. 16, §2.2.4 · pp. 18–19

2.2.2 Brief Overview of Different Numerical Methods

  • FDMs are simple and converge quickly but rely on regular meshes, FEMs accommodate irregular grids, and SEMs combine geometric flexibility with spectral accuracy for heterogeneous wavefield simulation.SEMs use hexahedral elements and high-degree Lagrangian interpolants; Gauss-Lobatto-Legendre integration yields a diagonal mass matrix and permits explicit methods without linear-system inversion.Source: §2.2.2 · pp. 16–17

2.2.3 General Limitations and Applicability to Seismology

  • Numerical wave solvers require discretization and face stability, meshing, boundary-condition, and computational-cost challenges, especially for high-resolution simulations in heterogeneous media.Explicit schemes must satisfy the CFL condition; finer spatial resolution also requires smaller time steps. Absorbing boundaries such as PMLs are complex to implement.Source: §2.2.3 · pp. 17–18

Traditional Numerical Methods

  • Machine learning has been applied to seismic signal analysis and simulation, but data-driven models can be difficult to interpret and may generalize poorly beyond training data; PINNs incorporate physical laws and can work with minimal or no data.PINNs impose physical laws, usually expressed as ODEs or PDEs, by penalizing violations in the loss function.Source: Traditional Numerical Methods · pp. 19–21

2.3.2 Previous Work

  • PINN-like neural methods appeared in the 1990s [57], while Raissi, Perdikaris, and Karniadakis’ 2019 work established modern PINNs for forward and inverse PDE problems.Later work addressed PINN failures such as numerical stiffness and unbalanced back-propagation with learning-rate annealing, alongside specialized architectures and training schemes [104] [109] [115].Source: §2.3.2 · pp. 22–23

Structuring the PINN Loss Function

  • PINNs approximate an unknown PDE solution with a neural network and train it to satisfy governing equations and initial and boundary conditions, adding data loss when needed for inverse problems.The relative weights of physics, initial-condition, boundary-condition, and data losses may be trainable; data loss is not required for forward simulation.Source: Underlying PDE · p. 23, Structuring the PINN Loss Function · p. 24
  • PINN losses constrain the solution using PDE residuals, initial and boundary conditions, and available measurements, with training points typically sampled randomly or by Sobol sequences.Source: Structuring the PINN Loss Function · p. 24

Advantages of Unsupervised Learning in PINNs

  • For forward problems, PINNs can learn from a PDE residual without labeled data, and their random space-time sampling avoids explicit discretization.They are therefore categorized as unsupervised for forward problems; inverse problems are categorized as supervised.Source: Advantages of Unsupervised Learning in… · p. 25

Designing the Neural Network Architecture

  • Fully connected feed-forward networks are the most commonly adopted PINN architecture, using layerwise weights, biases, and activations to map inputs to outputs.Activation functions introduce nonlinearities that enable modeling complex functions; tanh is often preferred to ReLU for PINN training.Source: Designing the Neural Network… · pp. 25–26

Training Process and Backpropagation

  • Training uses backpropagation to compute loss gradients with respect to network parameters and update them toward minimizing the loss.The process propagates inputs forward, evaluates the loss, applies the chain rule to its components, and updates parameters using the gradient.Source: Training Process and Backpropagation · p. 27

Optimisation Algorithms in PINN Training

  • ADAM adapts parameter learning rates using first and second gradient moments, while LBFGS approximates inverse-Hessian curvature and can converge in fewer steps but uses more memory.LBFGS is poorly suited to mini-batches and may be unstable when gradient-computation data points change; ADAM supports mini-batch training. An ADAM-then-LBFGS strategy depends on ADAM reaching an accurate preliminary approximation.Source: Optimisation Algorithms in PINN Training · pp. 27–28

Derivative Calculation and Implications for Activation Functions

  • Automatic differentiation computes PDE-required input derivatives, and activation functions for a PDE with k-th order derivatives should belong to Ck.ReLU is continuous but not differentiable at zero, whereas tanh is C∞ and supports smooth differentiation.Source: Derivative Calculation and… · pp. 28–29

PINNs

This section assesses baseline PINNs for the elastic wave equation in seismology, describing model and training choices and examining their performance across constant and heterogeneous settings. It also outlines limitations involving accuracy, high-frequency features, tuning, training, and retraining for new source locations, alongside fast single-time inference.

3.1 Introduction

  • PINNs offer meshfree continuous solutions and can be conditioned on problem parameters for fast surrogates, but the baseline has underwhelming accuracy in challenging setups and requires retraining for new scenarios.These limitations motivate later architectural and source-conditioning strategies.Source: §3.1 · pp. 31–32

3.2 Contributions

  • The study tests a standard feedforward PINN for a single seismic source across constant, smoothly heterogeneous, and layered parameter models, and varies source size to probe higher-frequency behavior.The broader assessment asks whether PINNs can serve as feasible alternatives across increasingly complex settings.Source: §3.1 · p. 31, §3.2 · p. 32

3.3 Related Work

  • Prior elastic-wave PINN studies covered homogeneous domains [89] or scattered frequency-domain problems, whereas this work uses time-domain modeling and examines heterogeneous models and source sizes.The literature review identifies elastic-wave applications as less explored than acoustic-wave applications.Source: §3.3 · pp. 33–34

3.4 Methods

  • The baseline maps spatial and temporal coordinates to two displacement components with a feedforward network, and its physical loss uses the PDE residual.Automatic differentiation computes the second-order space and time derivatives used in the loss.Source: §3.4.1 · pp. 34–35, §3.4.2 · p. 35

3.4.3 Initial Condition

  • A hard-constraint solution Ansatz enforces the initial condition exactly at t = 0, with t1 = 0.1 found empirically optimal; too small a t1 can slow convergence.The Ansatz blends the initial condition with the network prediction as time progresses.Source: §3.4.3 · pp. 37–38

3.4.5 Overall Loss

  • The smooth PINN output is treated as providing absorbing boundary behavior without explicit boundary conditions, while the overall loss is reduced to the physics-based term LP without labeled data.The study focuses on forward problems and uses an analytically defined initial condition.Source: §3.4.4 · p. 39, §3.4.5 · p. 39

3.4.6 Sampling

  • Sobol low-discrepancy sampling supplies collocation points; about 100,000 points sufficed in the specified homogeneous domain, while more complex geometries may require more.The exact number used varies by experiment.Source: §3.4.6 · p. 40

3.4.8 Training Setup

  • Keeping time inputs in [0, 1] did not hinder convergence or accuracy, and LBFGS was selected after it outperformed standalone ADAM and sequential ADAM-LBFGS in convergence.Spatial inputs were scaled to [−1, 1] to match the tanh activation range.Source: §3.4.7 · pp. 40–41, §3.4.8 · p. 41

3.4.8 Training Setup; 3.4.9 Workflow and Test Loss Evaluation

  • Test accuracy was the average relative L2 error across 100 time steps against a DEVITO FDM reference, while training used full-batch LBFGS.Access to GPUs with up to 80 GB of memory enabled experiments with more than one million collocation points; the test simulation used second-order time and fourth-order space discretization.Source: §3.4.9 · p. 42, §3.4.8 · p. 42

3.5.1 Constant Parameters; 3.5.2 Heterogeneous Parameter Models

  • Average relative L2 error was 2.22% for constant parameters, 2.88% for the mixture model, and 5.02% for the layered model, reaching 13% at t = 0.99.The layered PINN captured primary reflections and refractions, but struggled with low-frequency wavefront features late in the simulation; the constant-case error may be overstated because DEVITO is a non-analytical reference with discretization error.Source: §3.5.1 · p. 44, Figure 3.5 · p. 45, §3.5.2 · pp. 50–51

3.5.1 Constant Parameters

  • Fewer than 32 neurons per layer or two hidden layers failed to converge adequately; five layers with about 128 neurons gave the best observed architecture performance.Further increases in width or depth did not significantly improve accuracy; the practical tuning ranges were 4–6 layers and 64–200 neurons.Source: Figure 3.6 · p. 47, §3.5.1 · p. 47
  • Sources with σ ≤ 0.06 hampered convergence, while error declined quasi-quadratically as σ increased up to 0.16 and then stopped improving.The small-source difficulty was attributed to spectral bias [17][92] and violation of the Nyquist-Shannon sampling criterion; the behavior beyond σ = 0.16 remained unexplained.Source: §3.5.1 · p. 49

3.5.2 Heterogeneous Parameter Models

  • The best t1 differed by setting: 0.1 for constant parameters and 0.2 for the more complex mixture and layered cases.The findings indicate that small changes in t1 can substantially alter error, so each experiment requires its own hyperparameter tuning.Source: Figure 3.11 · p. 53, §3.5.2 · p. 53

3.5.3 Efficiency

  • PINNs were faster for a single-time inference, but DEVITO was markedly faster when producing high-resolution fields at all 100 time steps, and PINN training took hours.Training and FDM execution were performed on different hardware; early stopping or domain decomposition were suggested as ways to reduce training time.Source: Table 3.1 · p. 53, §3.5.3 · p. 54

3.6 Discussion

  • The discussion identifies spectral bias in complex or high-frequency cases, inconsistent and costly hyperparameter tuning, and hours-long training with retraining needed for new sources or parameters as practical limitations.The authors state that PINN accuracy had not surpassed traditional numerical methods.Source: §3.6 · p. 55, Efficiency · p. 55

3.7 Conclusion

  • Across constant, highly heterogeneous mixture, and layered settings, PINNs adapted to increasing model complexity and learned primary solution components, but faced significant accuracy challenges in layered models.The conclusion also reports that decreasing source size challenges convergence.Source: §3.7 · p. 55
  • Retraining for every new source location makes PINNs practically unviable; the authors aim to improve accuracy and develop a versatile model that avoids retraining.Source: §3.7 · p. 56

Neural Network Architecture Search … 4.4 Methods

This section searches a spectrum of neural-network architectures for incorporating wave physics into PINNs, describing the motivation, related work, basis functions, and specialized design templates. It also examines amplitude and encoder-decoder designs, convergence failures, and the models’ relative prior-knowledge and restrictiveness, and reports an architecture that outperforms alternatives across multiple elastic-wave settings.

4.1 Introduction

  • The study tests whether increasing physical prior knowledge—from an unconstrained fully connected PINN to specialized networks—improves accuracy and convergence while narrowing the solution space.The motivating wave properties include oscillation and outward propagation; heterogeneous media also produce P- and S-wave polarizations, with dispersion and attenuation.Source: §4.1 · pp. 57–58

4.2 Contributions

  • The chapter identifies an architecture that outperforms all other architectures, including the baseline PINN, across Lamé parameter settings and source sizes, and also applies it to the acoustic wave equation.The contribution passage presents the acoustic-wave application as evidence of generalizability.Source: §4.2 · p. 59

4.3 Related Work

  • The authors say no prior study directly matches their multi-design search embedding wave-physics knowledge in seismology, although related work includes specialized task-specific networks,, [119].They also cite Gabor-basis PINNs for the Helmholtz equation [5] and prior uses of wavelets, Fourier features, and plane waves.Source: §4.3 · p. 60

4.4 Methods; 4.4.1 Basis Functions; 4.4.2 Design Templates

  • The methods examine four design templates using sinusoidal functions, wavelets, plane waves, and Bessel functions, and also discuss designs that prove unsuccessful in practice.The chapter places the models along the prior-knowledge spectrum.Source: §4.4 · p. 61, §4.4.1 · p. 61, §4.4.2 · pp. 62–63

4.4.3 PINNs- tanh

  • The default PINN-tanh is the spectrum’s unconstrained baseline, independent of prior wave knowledge and used to compare architectures with greater complexity or domain-specific insight.It is positioned at the extreme left of the spectrum.Source: §4.4.3 · p. 63

4.4.4 Input Transformation PINNs

  • Input-transformation PINNs map (t, x, y) into a higher-dimensional wavelet, plane-wave, or sinusoidal space before standard fully connected layers process it.The mapping layer has trainable parameters; wavelet mappings, for example, align with local oscillations and outward propagation.Source: §4.4.4 · pp. 63–64

4.4.5 Encoder-PINN

  • Encoder-PINNs use an FCN to generate input-dependent wavelet or plane-wave parameters, then combine the resulting basis functions linearly to produce the wavefield.The authors state that, to their knowledge, prior work had not modeled the solution as a linear combination of custom basis functions that are themselves learned and input-dependent [5].Source: §4.4.5 · pp. 66–67

4.4.6 Amplitude-PINN

  • Amplitude-PINNs make only the wavelet or plane-wave amplitudes input-dependent through an FCN; the other basis parameters remain trainable but input-independent.The FCN-generated amplitude multiplies each custom basis function before the linear output layer.Source: §4.4.6 · pp. 68–69
  • Amplitude-PINNs multiply an encoder-produced amplitude by a custom wavelet or plane-wave layer that also receives the original inputs, without additional nonlinear layers.The custom layer has trainable but input-independent parameters.Source: Figure 4.5 · p. 70

4.4.7 Encoder-Decoder-PINN

  • Encoder-decoder PINNs add a secondary FCN after the wavelet or plane-wave layer, enabling nonlinear superposition and input-dependent parameters that adapt without predefined or random values.The decoder may help model complex features such as reflections and refractions in layered media.Source: §4.4.7 · pp. 70, 72, Figure 4.6 · p. 72

4.4.8 Failure Cases

  • After testing a broad range of architectures, including models that remained unconverged despite rigorous hyperparameter optimization, the authors focus comparison on functional models.They note that configuring and testing more than ten architectures across initialization schemes and hyperparameter settings required significant time and resources.Source: §4.4.8 · p. 73, Modified-Radial-Bessel-PINNs · p. 75

Modified-Radial-Bessel-PINNs

  • MRB layers showed no convergence despite exhaustive tuning, and MFF-PINNs consistently failed; the illustrated single-neuron MRB wavefield used manually tweaked, not learned, parameters.The authors attribute MRB's failure to a loss landscape too complex for the optimizer; MFF-PINNs use seven trainable parameters per neuron.Source: Modified-Radial-Bessel-PINNs · pp. 73–74, Figure 4.7 · p. 74
  • Other unsuccessful designs included sine-activation PINNs, sinusoidal encoder variants, separate component-specific wave layers, and traditional FCNs nesting wavelets or plane waves.The authors link nested wave-layer failures to physical incongruity: wavelets should take modified Cartesian-space inputs.Source: Modified-Radial-Bessel-PINNs · pp. 74–75

4.4.9 Placement on the Spectrum

  • The authors’ spectrum is subjective: PINN-tanh is least informed and restrictive, while models without decoders and specialized direct-output designs are more restrictive.Encoder-decoder models allow nonlinear superposition, encoder-only models form linear combinations, and amplitude designs are the most restrictive of the four templates discussed.Source: §4.4.9 · pp. 75, 77

4.5 Results · 4.6 Discussion · 4.7 Conclusion

The section assesses encoder–decoder PINNs with wave-physics-informed layers across elastic and acoustic wave-equation settings, including how architectural constraints affect accuracy, convergence, and robustness. It also notes that retraining for each new problem limits efficiency.

4.5.1 Evaluation along the Spectrum

  • Accuracy improves toward encoder-decoder PINNs, which achieve the lowest errors, but overly restrictive designs can worsen accuracy or fail to converge.The wED and pED variants remain accurate across experiments; MFF and MRB PINNs are excluded from the figures because training produced no meaningful loss reduction.Source: §4.5.1 · pp. 79, 81
  • Average relative L2 error falls from 3.34% to 1.56% across the three experiments, and appropriate encoder-decoder PINNs consistently achieve below 1.46%.The comparison is against the FDM solution produced using DEVTIO.Source: Table 4.1 · p. 82, §4.5.1 · p. 82

Constant Parameters

  • Relative L2 errors were 1.28% for wED-PINN with constant parameters, 1.39% for wED-PINN with mixture parameters, and 1.46% for pED-PINN with layered parameters.In the layered case, pED-PINN’s relative L2 error was approximately three and a half times lower than the standard model’s.Source: Constant Parameters · pp. 83, 85, 87
  • For the first novel source (initial uy displacement zero; t1=0.1), wED-PINN’s relative L2 error was 1.93% versus PINN-tanh’s 5.36%; for the second source (t1=0.2), errors were 1.4% versus 2.36%.Source: Constant Parameters · pp. 89, 91
  • With a smaller source in the high-frequency layered case, PINN-tanh error approached 15%, while pED-PINN achieved 2.94% without comparable severe deficiencies.ED-type PINNs show reduced sensitivity across t1 values [0.07, 0.1, 0.2], and wED-PINN has lower L2 errors across tested source sizes; error generally falls as source size increases, except at σ = 0.08.Source: Constant Parameters · pp. 85, 87, 91, Figure 4.18 · p. 93

4.5.2 Encoder-Decoder-PINNs: Extended Validation

  • Across every tested scenario, encoder-decoder PINNs outperform the PINN-tanh baseline in relative L2 error.The comparison spans constant, mixture, and layered Lamé parameters, novel seismic sources, and a high-frequency layered case.Source: Table 4.2 · p. 94, Constant Parameters · p. 94

Constant Parameters

  • PINN-tanh performs acceptably for constant acoustic velocity but struggles to converge in complex cases, underscoring the need for targeted adjustments to standard PINNs.The study did not exhaustively tune the acoustic models; prior reported results for complex cases typically used tailored hyperparameter optimization and curriculum-based training.Source: Constant Parameters · pp. 95–97
  • Errors below 1% in the constant-velocity acoustic case, with commendable accuracy in mixture and layered cases, show wED-PINN’s robustness across the tested scenarios.The study used the previously selected wED-PINN configuration without exhaustive acoustic hyperparameter search; PINN-tanh configurations were tuned separately for the three scenarios.Source: Constant Parameters · pp. 95–97, Table 4.3 · p. 97

4.6 Discussion

  • Encoder–decoder PINNs occupy a middle ground between minimal and excessive physics constraints, outperforming standard PINNs across complex parameters, multiple sources, and acoustic settings.Their balance combines physical knowledge with sufficient flexibility, and their acoustic generalization required no extensive hyperparameter optimization.Source: §4.6 · p. 99
  • Architectures aligned with physical principles offer a more scalable, flexible approach than scenario-specific solutions, improving accuracy and reducing intensive hyperparameter optimization.The discussion identifies better resource utilization as a benefit, while the conclusion cautions that overall PINN efficiency is not substantially improved.Source: §4.6 · p. 99, §4.7 · pp. 99–100

4.7 Conclusion

  • The improved architectures do not resolve PINNs’ efficiency challenge: they must still be retrained for each new problem setting.The conclusion says accuracy improves and hyperparameter-search burden is reduced to a lesser extent, but efficiency remains for future work.Source: §4.7 · pp. 99–100

Conditioning

This section examines conditioning PINNs on source location to support predictions across multiple locations without retraining, for elastic and acoustic wave equations, including highly heterogeneous settings. Reported comparisons assess accuracy against standard PINNs, while a many-source case study considers potential efficiency advantages alongside accuracy and training-time limitations.

5.1 Introduction

  • Conditioning addresses the need to retrain for every source location, with the aim of predicting wavefields for many locations without repeated PINN training or FDM simulation.The motivation includes broad-band regional simulations of potential earthquake-induced surface responses under uncertainty in seismic origin.Source: §5.1 · pp. 101–102

5.3 Related Work

  • The approach differs from cited source-conditioned PINN studies by targeting the time-domain elastic wave equation rather than the acoustic equation or frequency-domain elastic equation.The section also distinguishes its no-labelled-data approach from neural operators and PINOs, which rely on training data.Source: §5.3 · pp. 103–104

5.4 Methods

  • The conditioned network extends its input mapping from R3 to R5 by adding source coordinates, without changing the loss functions, and adjusts the initial condition to center it on the source.The method uses no training data and tests an encoder-decoder PINN and a standard PINN with heterogeneous Lamé parameters.Source: §5.2 · p. 102, §5.4 · p. 104

5.4.2 Sampling Strategies

  • Split-conditioning, especially with Ns < 100, tended to cause overfitting and suboptimal local minima; full-conditioning generally converged and interpolated better, so the study used it exclusively.Source locations were sampled only in Ωs; expanding to the full domain would require 6.25 times as many collocation points to maintain source density and was infeasible under GPU memory constraints.Source: §5.4.2 · pp. 105–106

5.4.3 Training Cost and Time

  • Training was constrained to about half a million collocation points and could take up to 30 hours per conditioned PINN.Full-batch LBFGS training on shared 80 GB NVIDIA A100 GPUs limited the feasible point count, largely because of second-order derivatives.Source: §5.4.3 · p. 106

5.5.1 Elastic Conditioning

  • 4.08% versus 5.45% mean relative L2 error: the pED-PINN outperformed PINN-tanh across the elastic source subdomain.In the central subdomain, the errors were 3.79% and 5.15%, respectively; errors tended to peak near boundaries and corners, where nearby source locations were less dense.Source: §5.5.1 · pp. 107, 109, Figure 5.3 · p. 110

5.5.2 Acoustic Conditioning

  • 2.21% versus 4.70% and 4.84% versus 10.33% average relative L2 error: pED-PINN outperformed PINN-tanh for two acoustic source locations.The examples correspond to s1 = (0.15, −0.05) and s2 = (−0.3, −0.05), respectively.Source: Figure 5.8 · p. 115, §5.5.2 · p. 115, Figure 5.9 · p. 116

5.5.3 Consolidated Results

  • Table 5.1 consolidates mean average relative L2 errors and their standard deviations for both models, subdomains, and wave equations.The supplied passage describes the table's scope but does not provide its cell values.Source: §5.5.3 · p. 116
  • The encoder-decoder model incorporating prior wave physics successfully conditioned on source location for elastic and acoustic equations in highly heterogeneous settings, with reasonably low errors but room for improvement.The network accurately simulated wave propagation in complex settings.Source: Table 5.1 · p. 116, §5.5.3 · pp. 116–117

5.6.1 Introduction

  • Conditioning lets a PINN infer a wavefield directly at a specified time and evaluate multiple source locations concurrently, but retraining for each new scenario can negate the speed advantage.Traditional methods require separate simulations for each source, whereas a conditioned PINN can represent source location, spatial coordinates, and time in one learned function.Source: §5.6.1 · pp. 117–118

5.6.2 Setup

  • For hazard assessment, conditioned PINNs can process receiver, source, and time inputs in one forward pass to compute peak displacement amplitudes across candidate earthquake sources.The setup uses 30 receivers and 10 time points per source-receiver pair; traditional FD/FEM/SEM approaches require an individual simulation for each source.Source: §5.6.2 · pp. 118–120

5.6.3 Results

  • Against a 512×512 pseudo-reference, 64×64 FDM averaged approximately 7% relative L2 error, versus PINN errors of 4.1% and 3.8% by subdomain; 256×256 FDM averaged approximately 1%.For 100,000 sources, FDM could take days or weeks, while PINN inference took minutes on CPU and seconds on GPU; the most complex PINN training took up to 30 hours, and efficiency depended on source count and resolution.Source: §5.6.3 · pp. 120–121, Table 5.2 · p. 121

5.7 Discussion

  • The pED-PINN improved accuracy over the standard PINN in the conditioned heterogeneous setting, while added collocation points or a better conditioned wavelet-based architecture were proposed as routes to further gains.Using more points would require mini-batch training and compatible optimizers because GPU memory was fully used; wED-PINN performed significantly worse than pED-PINN when conditioned, and a combined wavelet-plane-wave model was suggested.Source: §5.7 · pp. 122–123

5.8 Conclusion

  • Conditioning addresses PINN efficiency by avoiding retraining for each source and can enable rapid hazard assessment, although accuracy can fall in some cases and training can be lengthy.The chapter reports speed advantages over traditional methods for specific applications, not universal superiority across seismological analyses.Source: §5.8 · p. 123

Conclusion

The discussion finds that encoder-decoder PINNs with wavelet or plane-wave layers improve accuracy, while source conditioning supports faster inference in some scenarios. Future work targets training constraints, broader inputs and domains, and more expressive architectures.

6.1 Discussion

  • An intermediate level of physical knowledge integration achieved optimal PINN performance without overly constraining the solution space; encoder-decoder designs with wavelet or plane-wave layers outperformed conventional PINNs in accuracy across seismic scenarios and PDEs.Source: §6.1 · p. 125
  • Source-location conditioning improved efficiency despite extended training times, and the conditioned encoder-decoder PINN was more accurate than conventional PINNs and faster at inference than traditional methods in certain scenarios.In a seismic hazard case study, it produced assessments for many potential sources significantly faster than traditional solvers.Source: §6.1 · p. 125

6.2 Future Work

  • LBFGS is incompatible with batched training, limiting collocation points to available GPU memory; investigating batched-compatible ADAM could permit more points and improve accuracy, especially in conditioned cases.Source: §6.2 · p. 125
  • Combining plane-wave and wavelet layers and explicitly encoding source location in those layers may improve accuracy, since each architecture excels under different conditions.Source: §6.2 · pp. 125–126
  • Conditioning on source size, type, and parameter models could expand applicability; PINOs [62] may manage this larger input space, while pretraining and task-specific fine-tuning could help but make the method more data-driven.Source: §6.2 · p. 126
  • Combining the encoder-decoder architecture with domain decomposition could localize wave layers to subdomains and shorten training; large-scale real-world geological models are still needed to assess practical applicability.Source: §6.2 · p. 126
  • Testing advanced PINNs on coupled elastic-acoustic domains could clarify their performance for solid-fluid wave interactions and possible applications in planetary science and terrestrial hazard prediction.Source: §6.2 · pp. 126–127

Background · A.1 Python Implementation of DEVITO

The appendix gives the DEVITO numerical baseline implementation, initializing displacement fields, forming elastic-wave equations, and executing forward-time stencils with boundary conditions.

A.1 Python Implementation of DEVITO

  • The baseline assigns the initial x- and y-displacement fields to time indices 0 and 1.The listing sets ux.data[0] and ux.data[1] to u0x, and uy.data[0] and uy.data[1] to u0y.Source: §A.1 · p. 143
  • The x-component equation uses density-weighted acceleration minus the x-divergence of stress.The code defines pde_x = rho_solid * ux.dt2 - div_stress_ux.Source: §A.1 · p. 143
  • The y-component equation uses density-weighted acceleration minus the y-divergence of stress.The code defines pde_y = rho_solid * uy.dt2 - div_stress_uy.Source: §A.1 · p. 143
  • DEVITO forward stencils are formed by solving each component’s PDE for its forward displacement field.The listing creates stencil_x and stencil_y using Eq and solve.Source: §A.1 · pp. 143–144
  • The operator combines both component stencils with boundary conditions and runs them to end_time using dt.The listing constructs Operator([stencil_x] + [stencil_y] + bc) and calls op(time=end_time, dt=dt).Source: §A.1 · p. 144
  • The listing is identified as the Python implementation of the underlying numerical baseline built with DEVITO.Source: §A.1 · p. 144

PINNs · B.1 Pytorch Implementation of Physical Loss

The physical-loss implementation uses PyTorch automatic differentiation to compute derivatives of the approximated solution, then forms strain and stress components from those derivatives.

B.1 Pytorch Implementation of Physical Loss

  • The implementation evaluates the approximate solution at input_s and differentiates its first two output columns with respect to the inputs.It uses torch.autograd.grad with create_graph=True for both first derivatives.Source: §B.1 · p. 145
  • It computes second derivatives of both output components along the first input coordinate.The resulting tensors are assigned to dt2_x and dt2_y.Source: §B.1 · p. 145
  • The strain tensor’s first diagonal entry is gradient_x[:,1], and its second diagonal entry is gradient_y[:,2].Both entries are constructed within the symmetrized strain tensor calculation.Source: §B.1 · p. 145
  • The off-diagonal strain entries are one-half the sum of gradient_x[:,2] and gradient_y[:,1].The code places this same sum in both off-diagonal tensor positions.Source: §B.1 · p. 145
  • The stress_tensor_00 component combines lambda_m times the sum of the two strain diagonal entries with 2.0 times mu_m times eps[0,0].Source: §B.1 · p. 145
  • The off-diagonal stress component is computed as 2.0 times mu_m times eps[0,1].The code assigns this value to stress_tensor_off_diag.Source: §B.1 · p. 145

B.2. Minimum source width for Nyquist sampling

The supplied excerpt describes the baseline PINN physical-loss implementation, which computes a solid residual from inertia and stress divergence and takes its mean squared magnitude as the loss.

B.2. Minimum source width for Nyquist sampling

  • The baseline PINN computes the solid residual as density-scaled second time derivatives minus stress divergence, then defines the loss as the mean squared residual magnitude.The excerpt labels this implementation as the physical loss used in the baseline PINN.Source: §B.2 · p. 146

B.2 Estimation of Minimum σ to Comply with Nyquist–Shannon Sampling Theorem

The section estimates a conservative minimum Gaussian-source width of σ=0.06 to comply with Nyquist–Shannon sampling.

B.2 Estimation of Minimum σ to Comply with Nyquist–Shannon Sampling Theorem

  • The estimate uses the Nyquist spatial-sampling requirement, with Δx related to collocation-point count and domain length, and λmin related to fmax determined by σ and vmax.Source: §B.2 · pp. 146–147
  • Using N=100000, L=2, and vmax=0.89 (Vp), the estimated threshold near σ=0.045 is conservatively raised to σ=0.06 because of approximations.The adjustment is presented as a conservative estimate of the smallest source size that complies with the Nyquist–Shannon theorem.Source: §B.2 · p. 147

Neural Network Architecture Search · C.1 Modified Radial Bessel PINNs

This subsection specifies a modulated radial-Gaussian envelope and an approximation of the Bessel function of the first kind for the modified radial Bessel PINN. It identifies trainable and fixed parameters used in those definitions.

C.1 Modified Radial Bessel PINNs

  • The function g(x, y) defines the modulated radial-Gaussian envelope used in Equation (4.10).The supplied passage introduces its definition but does not include the complete formula.Source: §C.1 · p. 148
  • The envelope parameters αg, κ, γ, and ω are trainable.Source: §C.1 · p. 148
  • The Bessel function of the first kind is approximated, with constants λ0 = 0.0815, p1 = −0.2622, q = 0.5701, p̃1 = −0.3945, p0 = 0.5948, p2 = −0.0493, p̃0 = 0.4544, and p̃2 = 0.0327.The supplied passages show only fragments of the approximation formula.Source: §C.1 · p. 148

C.2 Far Field PINNs

The Far Field PINN adapts the approach in from three dimensions to two and models only the far field. Its neuron uses source-relative directions and trainable parameters to define the P- and S-wave radiation and displacement outputs.

C.2 Far Field PINNs

  • The Far Field neuron adapts to two dimensions and omits intermediate- and near-field equations, focusing solely on the far field.Source: §C.2 · pp. 148–149
  • Source-relative unit vectors define the directions originating at the seismic source location.Source: §C.2 · p. 149
  • Trainable parameters include characteristic time T, seismic moment M0, radiation-pattern parameters [θ1, θ2, θ3, θ4], and delay δ; the outputs are normalized.The neuron calculates x and y displacement components using P- and S-wave radiation patterns; wave speeds vp and vs are calculated as described in Equation (2.3).Source: §C.2 · p. 149

C.3 Model Hyperparameters

This section lists the model configurations used to generate the results in Table 4.1, with hyperparameter tables for constant, mixture, and layered Lamé-parameter models.

C.3 Model Hyperparameters

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