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Physics-informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions
Majid Rasht-Behesht, Christian Huber, Khemraj Shukla, George Em Karniadakis
TL;DR
The paper addresses wave propagation and full waveform inversion when seismic data are limited. It applies physics-informed neural networks to the 2D acoustic wave equation and reports accurate forward and inversion results across synthetic cases, while identifying practical limitations in training and scaling.
Problem
Seismic inversion and wave modeling must operate with limited spatial observations, motivating methods that rely less on labeled wavefield data.
Method
PINNs solve the 2D acoustic wave equation by optimizing a neural network against governing physics, early-time snapshots, boundary constraints, and observed seismograms.
Results
PINNs accurately solve forward propagation and recover wavespeed and wavefields across the considered synthetic inversion cases, including multiple seismic events.
Takeaways & Limitations
The meshless PINN framework flexibly handles diverse constraints, multiple data types, and multiple events with limited additional computational cost.
Takeaways & Limitations
PINNs require heuristic loss-weight and network-size choices, and large domains incur substantial memory costs requiring efficient GPU or domain-decomposition strategies.
Abstract
from arXiv · showhide
We propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called Physics-Informed Neural Networks (PINNs). In this study, we present an algorithm for PINNs applied to the 2D acoustic wave equation and test the model with both forward wave propagation and FWIs case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs formulation. We find that the current state-of-the-art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., magnetotellurics, gravity) as well as joint inversions because of its robust framework and simple implementation.
Acoustic Wave Propagation
The 2D acoustic wave equation describes propagation through a medium using a scalar wave potential, with wavespeed determined by material properties. Displacement is recovered from the potential gradient, while early-time snapshots and boundary constraints support forward modeling.
- The 2D acoustic wave equation models wave propagation in a medium with negligible density variations and no body forces.
- The scalar wave potential ϕ is governed by the Laplacian, source term f, and acoustic wavespeed α.
- External forces are set to zero and represented through a perturbation of the initial field at an early time.
- The displacement field is obtained as the gradient of the wave potential: (u_x, u_z) = ∇ϕ.
- Forward simulation uses boundary constraints, two early-time wavefield snapshots, and known spatial material properties; the snapshots constrain source shape and propagation direction.
Neural Network (NN)
The PINN uses a fully connected feed-forward network whose inputs are spatial coordinates and time, and whose output approximates the scalar acoustic wave potential. Other physical variables are derived through automatic differentiation.
- The network takes x, z, and t as inputs and outputs the scalar acoustic wave potential ϕ.
- Displacement and pressure are obtained by automatically differentiating the wave-potential network.
- Each hidden layer applies a linear transformation followed by a neuron-wise nonlinear activation function σ.
- The study uses tanh or sin activations in hidden layers and an identity function for the final output.
PINNs for the forward problem
For forward wave propagation, PINNs train a neural network to satisfy the acoustic PDE together with early-time snapshots and boundary constraints at randomly sampled points. The formulation casts wave simulation as loss minimization and accommodates free-surface and observed-data terms.
- PINNs for the forward problem: PINNs approximate the acoustic wave potential by enforcing the PDE, early-time snapshots, and boundary data at randomly selected training points.
- PINNs for the forward problem: The free-surface residual imposes the physical constraint at the model top, while the observed-data residual compares predicted and measured wavefields.
- PINNs for the forward problem: Two early-time snapshots can determine later wavefield evolution, and PINNs can use data at arbitrary times unlike traditional wave-equation solvers.
- PINNs for the forward problem: The loss combines PDE, snapshot, free-surface, and observed-data mean squared errors.
- PINNs for the forward problem: PDE training points are randomly sampled throughout the computational domain, while the schematic includes early snapshots, boundary data, and seismometer locations.
- PINNs for the forward problem: The forward and inverse problems are formulated as optimization over the network weights and biases.
- PINNs for the forward problem: Loss weights are selected heuristically, and disproportionate terms can delay convergence or produce the wrong solution.
PINNs for the seismic inverse problem
The inverse formulation estimates the spatial wavespeed distribution from sparse seismometer data, using an additional neural network for the medium properties. Including the free-surface constraint improves recovery of reflected phases.
- PINNs for the seismic inverse problem: The inverse problem estimates the spatial distribution of acoustic wavespeed from ground motion recorded at surfaces or boreholes.
- PINNs for the seismic inverse problem: A secondary neural network with independent weights and biases represents the medium’s wavespeed distribution.
- PINNs for the seismic inverse problem: Although seismometer data implicitly impose a free-surface condition, its explicit loss term is necessary for accurate recovery of reflected surface phases.
Normalization
PINNs require two normalization steps for convergence to the correct solution: mapping network variables to [−1,1] and scaling the acoustic wave PDE terms consistently.
- Map the network’s input and output variables to the interval [−1,1] ∈ℝ.
- Scale the acoustic wave PDE so both terms in equation 1 are on the same order.
- These two normalization steps are described as crucial for convergence to the correct solution.
Optimization
The PINN parameters are optimized with Adam using a learning rate of 1e-4, while training-data batch sizes are selected for each loss term and case study.
- Adam optimization uses a learning rate of 1e-4 and a suitably chosen batch size for each case study.Training stops when improvement becomes negligible or a fixed iteration limit is exceeded.
- Batch sizes are summarized separately for each loss term and case study.
- The PDE and free-surface conditions use variable training data, whereas early-time snapshots and seismograms use fixed datasets.The fixed datasets are used because variability in those data types is small enough for relatively small data sizes.
Validation
The forward PINN is tested in a heterogeneous medium against SpecFem2D solutions, using sparse early-time snapshots as training data and comparing predicted wavefields with ground truth.
- SpecFem2D solutions provide ground truth and generate the early-time snapshots and synthetic seismograms used for testing.The authors state that analytically or numerically generated snapshots can be used, with real seismograms intended for future inversions.
- The heterogeneous forward case uses a 3 km/s background, a Gaussian perturbation reaching 2 km/s, and a 2.5 km width.
- The forward model uses 20,000 PDE-loss points and 3,600 points for each early-time snapshot.
- PINN predictions provide a good wavefield solution while using only two early-time snapshots as labeled training data.The PDE supplies the information used to simulate propagation through the heterogeneous region.
- Misfits are computed on a separate input dataset from training to avoid bias in the network parameters.
Application of PINNs to inverse modeling
PINNs are applied to synthetic full waveform inversions using early-time wavefield snapshots, sparse seismograms, and prescribed or unknown wavespeed structure. Across homogeneous, anomalous, and teleseismic cases, they recover wavefields and structures accurately while handling boundary conditions and multiple events, subject to smoothing and heuristic design choices.
- Inversion strategy: The inversion strategy uses SpecFem2D-generated seismograms and two early-time displacement snapshots as PINN training data.The snapshots constrain the seismic source and propagation direction, while the seismograms provide inversion observations.
- Homogeneous crosswell: In the homogeneous crosswell case, PINN accurately recovers wavespeed, wavefield, and synthetic seismograms despite a relatively poor initial wavespeed guess.The setup uses a 20 Hz Gaussian point source and 20 seismometers over a 0.4-second window.
- Boundary conditions: PINN matches free-surface reflected waves in amplitude, waveform, and timing without explicitly enforcing absorbing conditions in the loss function.The authors attribute automatic absorption to the smooth, infinitely differentiable PINN solution.
- Ellipsoidal anomaly: For a discontinuous ellipsoidal anomaly, PINN retrieves the anomaly’s location, dimensions, and magnitude, but the inverted wavespeed is smoother than the true discontinuity.The smoothed structure still produces an accurate later wavefield and excellent matches across 20 seismograms, with larger end-window discrepancies attributed to imperfect final-time optimization.
- Network size: A smaller network gives a good wavespeed estimate but a less accurate recovered wavefield, supporting the use of a deeper network for improved accuracy.The result indicates that network size affects wavefield fidelity even when inversion structure remains satisfactory.
- Multiple events: Multiple teleseismic events are superposed and modeled with one network, avoiding a separate network for each event and limiting added computational cost.The approach exploits linearity of the acoustic wave equation, while destructive interference at seismometers can require separate event groups and networks.
- Computational considerations: Computational cost depends strongly on source frequency and wavespeed complexity, with smaller networks sufficient for simpler inverse models.The study also reports that PINNs can use prior wavespeed constraints, although they perform well without an educated starting model.
- Limitations: PINN design remains constrained by heuristic selection of loss-term weights and network size, while large domains impose substantial GPU memory demands.The authors identify convergence criteria, PDE training-point density, and scalable domain decomposition as areas requiring further work.
Conclusion
The study concludes that PINNs can solve acoustic wave propagation and full waveform inversions with limited data by combining governing equations with flexible, meshless constraints. Multiple-event inversions can be performed with limited additional memory or computational cost.
- PINNs solve acoustic wave propagation and full waveform inversions by relying on governing partial differential laws when data are limited.
- Multiple-event inversions are performed with limited additional memory or computational cost.
- The method combines flexible constraint handling, meshless modeling, and simple formulation and implementation.
Appendix A
Appendix A specifies two normalization steps for the PINN formulation and provides access to a sample implementation. The variables are scaled to stabilize the neural-network representation and wavespeed parameterization.
- The PINN formulation uses two important normalization steps.
- Each neural-network input variable is mapped to the interval [−1,1].
- The wave PDE is scaled so that wavespeed is constrained to the interval [0,1].
- A sample Python script for the study is available through Brown University’s data repository.
Supporting information
Supporting experiments show that PINN inversion quality depends on network expressiveness and training duration, while smaller or shorter-trained models still recover the main structural features with increased boundary smearing.
- Network-size test: A 4-layer, 50-neuron network recovers the ellipsoidal anomaly’s location, approximate size, and strength despite an uneducated initial guess.The experiment uses a 20 Hz point source and retains the other Case 2 parameters.
- Network-size test: The smaller network produces greater anomaly-boundary smearing and slightly larger wavefield errors near the velocity-anomaly boundaries, while seismogram fit changes little.The wavefield comparison is shown in Figure S2 and the seismogram comparison in Figure S3.
- Training-duration test: After 70,000 epochs, the teleseismic inversion yields an acceptable wavespeed model but slightly more boundary smearing than the 400,000-epoch result.This shorter-training experiment uses the ellipsoidal anomaly with two teleseismic plane waves.
- Figure overview: The supporting figures compare ground truth, inverted or predicted wavefields, pointwise differences, and input versus predicted seismograms for the crosswell and teleseismic cases.Figures S1–S5 show models and wavefields; Figures S2 and S4–S5 include absolute or pointwise differences, while Figures S3 and S6 show seismograms.