Source-linked AI summary
Solving the wave equation with physics-informed deep learning
Ben Moseley, Andrew Markham, Tarje Nissen-Meyer
TL;DR
The paper asks whether PINNs can solve the wave equation despite its multi-scale, propagating and oscillatory solutions. It trains neural networks with the governing equation and boundary conditions, tests increasingly complex 2D acoustic media, and conditions one model on source location. The networks accurately simulate the tested wavefields, generalise beyond boundary training data and across source locations, and allow single-step queries of arbitrary space-time points after training.
Problem
The paper investigates whether PINNs can effectively solve the wave equation, whose multi-scale, propagating and oscillatory solutions create distinctive challenges.
Method
The method represents the wavefield with a neural network constrained during training by the governing equation and boundary conditions, using curriculum learning and source-location conditioning.
Results
PINNs accurately solve increasingly complex acoustic wavefields, model dynamics beyond boundary training data, and generalise across source locations without retraining.
Takeaways & Limitations
Once trained, the network can query arbitrary space-time points in a wavefield without computing the entire field, offering an efficient simulation approach within the tested scope.
Abstract
from arXiv · showhide
We investigate the use of Physics-Informed Neural Networks (PINNs) for solving the wave equation. Whilst PINNs have been successfully applied across many physical systems, the wave equation presents unique challenges due to the multi-scale, propagating and oscillatory nature of its solutions, and it is unclear how well they perform in this setting. We use a deep neural network to learn solutions of the wave equation, using the wave equation and a boundary condition as direct constraints in the loss function when training the network. We test the approach by solving the 2D acoustic wave equation for spatially-varying velocity models of increasing complexity, including homogeneous, layered and Earth-realistic models, and find the network is able to accurately simulate the wavefield across these cases. By using the physics constraint in the loss function the network is able to solve for the wavefield far outside of its boundary training data, offering a way to reduce the generalisation issues of existing deep learning approaches. We extend the approach for the Earth-realistic case by conditioning the network on the source location and find that it is able to generalise over this initial condition, removing the need to retrain the network for each solution. In contrast to traditional numerical simulation this approach is very efficient when computing arbitrary space-time points in the wavefield, as once trained the network carries out inference in a single step without needing to compute the entire wavefield. We discuss the potential applications, limitations and further research directions of this work.
I. INTRODUCTION
The wave equation is difficult for conventional and data-driven methods because of its complex, multi-scale dynamics and limited generalisation. This work evaluates PINNs for wave simulation, extending them with source conditioning and curriculum learning while addressing interface discontinuities.
- Motivation: Wave-equation simulation is important across physics, but large 3D numerical solutions can require billions of mesh points and costly computation.Discretisation error and fine grids further increase the computational burden for long-distance wave propagation.
- Motivation: Existing deep-learning simulators often perform poorly outside their training distribution because they rely primarily on training data.PINNs instead impose governing equations during training, offering a potential route to generalisation beyond boundary data.
- Motivation: PINNs face distinctive wave-equation challenges from multi-scale, propagating and oscillatory solutions, including uncertainty about their performance in this setting.The paper tests these challenges in acoustic media of increasing complexity, from homogeneous to Earth-realistic models.
- Contributions: The proposed PINN accurately solves pressure responses in complex 2D acoustic media and models transmission, reflection, velocity-driven wave changes and spherical-divergence attenuation.It predicts later-time solutions from only the first few timesteps of boundary training data.
- Contributions: Conditioning the network on source location enables generalisation across initial source locations without retraining for each solution.The paper presents this as more efficient for varying source locations than traditional finite-difference modelling.
- Contributions: A curriculum-learning strategy improves convergence, while the paper identifies media-interface discontinuities as a specific PINN challenge and discusses ways to address them.The contribution extends the original PINN approach with both training and modelling strategies tailored to wave equations.
A. Physics-informed neural networks
PINNs approximate physical-system solutions with neural networks trained under both boundary-data and governing-equation constraints. This formulation uses automatic differentiation to impose physics while retaining a direct functional approximation of the solution.
- The governing differential equation is imposed as a constraint during network training.
- The approach can describe conservation laws, equations of motion, fluid dynamics, and many wave equations.
- PINNs represent the solution u(t, x) with a neural network Λ(t, x; θ).
- The loss combines known boundary values with a physics loss evaluated at points sampled across the input space.The boundary term promotes a unique solution, while the physics term encourages compliance with the underlying equation.
- Automatic differentiation provides the input gradients needed to derive the physics loss and differentiate it with respect to network weights.
B. Solving the wave equation
The study applies PINNs to the 2D acoustic wave equation, whose complex, oscillatory dynamics create representational, boundary-condition, and computational challenges. It tests increasingly complex velocity models and conditions the network on source location to avoid retraining for every initial condition.
- The 2D acoustic wave equation models pressure response in a medium whose density and velocity may vary spatially.Under constant density and negligible source terms, it reduces to the canonical wave equation.
- Wave-equation solutions involve broadband, oscillatory, dispersive behavior, including reflection, transmission, interface effects, and spherical spreading.
- PINNs face uncertainty over representing these dynamics, enforcing sufficient initial-value constraints, and computing costly second-order gradients.The second-order-gradient cost may restrict the type and size of usable neural networks.
- The study tests homogeneous, layered, and Earth-realistic Marmousi velocity models, retraining the PINN for each velocity model.The models increase in complexity to test the limits of the approach.
- For the Earth-realistic case, conditioning on source location enables generalization across initial conditions without retraining for each one.
C. PINN design and training strategy
The PINN approximates the wavefield at arbitrary space-time points by combining boundary data with a physics loss, then uses staged training with an expanding time horizon. This design is tested against boundary-only training and supports prediction beyond the training interval.
- PINN design: The network takes arbitrary time-space points as input and outputs an approximation of the wavefield solution.The network uses a fully connected architecture and is trained for a fixed velocity medium.
- Loss construction: The loss combines boundary-data fitting with a physics term that penalizes violations of the wave equation.The physics term uses the fixed velocity model and a tunable balance parameter k.
- Conditioning: The source-location-conditioned network is trained across multiple source locations to generalise without retraining for each location.Source locations are included in the network inputs and sampled when evaluating the physics loss.
- Evaluation: The physics-informed network is evaluated against a boundary-only network using wavefield predictions and differential-operator values.The comparison is made for homogeneous and layered velocity models, with later-time plots outside the boundary-data interval.
- Training strategy: The physics loss is introduced halfway through training, while its time horizon expands progressively to aid convergence.The staged schedule first reconstructs the boundary data before learning wave propagation through time.
D. Boundary data generation
The experiments use three velocity models of increasing complexity and generate initial boundary data with finite-difference simulations. Common simulations use a fixed 2D grid, timestep, density, and source signature.
- Velocity models: Three velocity models test the PINN from simplest to more complex settings: homogeneous, horizontally layered, and Earth-realistic Marmousi.The homogeneous model has a fixed velocity of 2500 ms^-1, while the Marmousi model represents an Earth-realistic P-wave velocity field.
- Boundary-data generation: Finite-difference modelling generates the initial wavefield training data for each velocity-model case.The simulations provide the boundary condition used to train the PINN.
- Simulation setup: The simulations use a 300 × 300 grid with 5 m spacing, a 0.002 s timestep, constant density, and a 20 Hz Ricker source.For the first two models, a centered point source and the first 10 timesteps are used after the source term becomes negligible.
A. Homogeneous velocity model, single source
In the homogeneous case, the PINN accurately reproduces the finite-difference wavefield and generalises far beyond the short boundary-data interval. The layered case shows that the method captures complex propagation dynamics but struggles near velocity interfaces.
- Homogeneous velocity model: The homogeneous-model PINN accurately matches the finite-difference solution, including wavefront kinematics and spherical-divergence amplitude attenuation.A boundary-only network propagates the wavefront outside the training interval but has poor accuracy there.
- Layered velocity model: The layered-model PINN captures reflected and transmitted waves generated at velocity interfaces.These waves are absent from the initial wavefield, so the result tests propagation beyond the boundary data.
- Layered velocity model: The layered-model PINN also captures wavefield compression, expansion, and spherical-spreading attenuation.These effects arise as waves travel through regions with different velocities.
- Interface behaviour: Physics-loss values are higher near velocity interfaces, where the wavefield has discontinuous second-order gradients.The network learns sharp but not fully discontinuous contrasts, and errors remain along the interfaces.
C. Marmousi model, multiple sources
The PINN accurately models wavefield kinematics across the complex Marmousi model and generalises across unseen source locations without retraining. It remains efficient after training, but struggles with low-amplitude reflected waves and interfaces.
- Marmousi model: The Marmousi model includes faults, velocity inversions, and varied interface dips, with 20 unseen source locations used for testing.Boundary and physics losses converge more slowly than in homogeneous and layered models.
- Multiple sources: The PINN accurately models the initial wavefront and subsequent dynamics across source positions without retraining.This extends the approach to varying initial source locations.
- Limitations: The network struggles with low-amplitude secondary reflections, while interface modelling remains challenging because velocity discontinuities create second-order derivative changes.The authors suggest amplitude imbalance, finite-difference inaccuracies, and discontinuity handling as possible contributors.
- Training cost: Training the Marmousi PINN takes approximately one day on a single Nvidia Titan V GPU, with second-order derivative computation dominating the cost.The authors note that training-time optimisation was limited.
- Generalisation: Boundary data from 0.00–0.04 s supports predictions beyond 0.32 s, eight times the boundary-data range.Figure 7 compares predictions with finite-difference simulations for randomly selected test sources.
- Efficiency: After training, arbitrary source-position and space-time wavefield queries take about 0.001 s, versus about 1 s for full 2D finite-difference simulations.The reported comparison makes finite-difference modelling approximately 1000× slower per source simulation.
V. CONCLUSION
The study shows that physics-informed deep neural networks can solve the wave equation and generalise beyond boundary training data. Conditioning on source location enables predictions for multiple sources without retraining, while inference is efficient after training.
- Conclusion: PINNs solve the wave equation and generalise beyond their training set by incorporating physics constraints directly into the loss function.The approach is efficient after training and does not rely on discretisation.
- Conclusion: Conditioning the network on source location allows wavefield prediction across multiple source locations without retraining.This extends the method beyond a single initial condition.
Appendix A: Supplementary figures
The appendix provides supplementary comparisons for an unsmoothed layered velocity model and for training without curriculum learning. These tests examine how alternative training and model-smoothing choices affect PINN wavefield predictions.
- Figure A1: Figure A1 shows the layered velocity model without smoothing used in the supplementary comparison.The unsmoothed model is evaluated in Figure A2.
- Figure A2: Figure A2 compares PINN wavefield predictions with finite-difference ground truth for an unsmoothed velocity model and for training without curriculum learning.The latter uses the physics loss from the start and samples the full time range.