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Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks

Akshay Thakur, Matthew Zahr

arXiv:2610.02230v1cs.CEphysics.comp-phphysics.flu-dynstat.ML

TL;DR

Uniformly sampled PINNs can miss localized shocks because the residual is dominated by smooth regions. FT-PINN jointly trains a solution network and deformation map to concentrate collocation points at features. Across four shock problems, it resolves features where a matched vanilla PINN fails.

  • Problem

    Uniform collocation can undersample localized shocks while smooth regions dominate PINN residual optimization.

    Source: §1.1 · p. 2

  • Method

    FT-PINN jointly trains a solution network on a fixed reference domain and a parameterized deformation map against the pulled-back conservation-law residual.

    Source: §1.2 · p. 2, §6 · p. 18

  • Results

    Across four shock-dominated problems, FT-PINN resolved the features on the same collocation budget, architecture, and training horizon on which a vanilla PINN failed.

    Source: §6 · p. 18

  • Takeaways & Limitations

    The nonlinear deformation framework represents curved, oblique, or geometrically complex features without prior knowledge of their locations.

    Source: §1.2 · p. 2, §6 · p. 18

    The experiments solve viscous conservation laws; for Euler problems, artificial viscosity regularizes true discontinuities into smooth internal layers of width O(ν).

    Source: §5 · p. 9

Abstract

from arXiv · show

Physics-informed neural networks (PINNs) often converge to inaccurate solutions for conservation laws with shocks, because uniformly distributed collocation points undersample localized features and let the residual be dominated by regions that are already well resolved. We propose a feature-tracking PINN (FT-PINN) in which the solution network is defined on a fixed reference domain and composed with a diffeomorphic deformation map from a parameterized nonlinear manifold. The deformation and solution-network parameters are trained jointly by minimizing the pulled-back conservation-law residual. This lets collocation points concentrate along features of essentially arbitrary geometry, including curved, oblique, and merging shocks, without prior knowledge of their locations. The framework is agnostic to the choice of parameterization. Boundary preservation is enforced exactly through a tangential projection of the displacement, and folding is discouraged by a one-sided penalty on the Jacobian determinant. On four test problems (space-time viscous Burgers with merging shocks, a decelerating Burgers shock, the space-time Euler shock tube, and steady 2D Euler regular shock reflection), FT-PINN resolves shocks at their correct locations with a limited collocation budget. A vanilla PINN with the same architecture, budget, and training either misplaces the shocks or fails to form them.

1. Introduction

Uniform collocation can leave shock features undersampled while smooth regions dominate PINN optimization. The paper proposes a jointly trained, non-separable deformation map that clusters collocation points at features and tests the approach on four shock problems.

1.1. Resolving shocks using PINNs

  • Uniform collocation undersamples narrow shocks, letting well-resolved smooth regions dominate the residual and producing smeared, inaccurate, or missing features.For hyperbolic laws, the strong-form residual is invalid at a discontinuity; artificial viscosity instead regularizes the shock into a thin layer.Source: §1.1 · pp. 1–2
  • Existing clustering and domain-decomposition remedies require prior feature locations, while pointwise relocation incurs repeated optimization and lacks a coherent domain transformation.Earlier separable maps also restrict clustering to coordinate directions, limiting their ability to represent curved or oblique shocks.Source: §1.1 · p. 2

1.2. Proposed approach and contributions

  • FT-PINN jointly trains a non-separable deformation map and solution network so a uniform reference grid concentrates at features of arbitrary geometry without prior location knowledge.The formulation transplants the r-adaptive, shock-fitting paradigm into collocation and is agnostic to the map parameterization.Source: §1.2 · p. 2
  • A one-sided Jacobian-determinant barrier discourages map folding, while a tangential projector enforces boundary preservation exactly and permits redistribution along the boundary.The barrier does not oppose the volume contraction used to cluster points.Source: §1.2 · pp. 2–3
  • The framework is demonstrated on four shock-dominated problems, each compared with a vanilla PINN using identical architecture, training, and collocation budget.The comparisons also use an exact or high-resolution reference solution.Source: §1.2 · p. 3

2. Governing equations and discretization

The paper formulates viscous conservation laws in a general space-time setting, then describes their PINN approximation and a reference-domain transformation. The transformed equation is evaluated on a fixed reference domain after pulling back the solution and fluxes.

2.1. Viscous conservation laws

  • The governing formulation splits flux into inviscid and viscous parts and defines a strong-form residual whose pointwise values drive the physics-informed loss.The same conservation-law form admits unsteady problems when the domain is read as space-time, with divergence taken across spatial and temporal directions.Source: §2.1 · p. 3

2.2. Physics-informed neural network approximation

  • A PINN approximates the conserved state with a neural network and trains its weights against PDE and boundary-condition penalties using automatic differentiation.The solution network is a fully connected network with a random Fourier feature embedding.Source: §2.2 · p. 4

2.3. Reference-domain formulation

  • A parameterized diffeomorphism maps the reference domain to the physical domain, and pulling back the solution and fluxes yields an equivalent conservation law on the reference domain.A uniform reference grid therefore maps to a physical point distribution shaped by the deformation.Source: §2.3 · pp. 4–5

3. Feature-tracking PINNs via nonlinear deformation manifolds

FT-PINN jointly learns a solution on a fixed reference domain and a deformation that maps its collocation grid into physical space. The construction adapts points to features while preserving the domain boundary and discouraging map folding.

3.1. Nonlinear manifolds of deformations

  • A non-separable displacement field allows the deformation gradient to rotate and shear the grid, supporting alignment with curved and oblique features.The displacement must be smooth enough to support the second derivatives required by viscous fluxes.Source: §3.1 · p. 5

3.2. Radial basis function-based parameterization

  • The Gaussian RBF map starts at the identity and can be replaced by other smooth radial functions or an FCNN without changing the tracking construction.The learned map can fold or move points outside the domain, so additional constraints are needed.Source: §3.2 · pp. 5–6

3.3. Induced collocation distribution

  • Optimizing the deformation parameters maps a fixed uniform reference grid into an adaptive physical collocation distribution without per-point updates or resampling.The mapped set retains the reference grid's cardinality, and per-iteration cost scales linearly with the number of collocation points.Source: §3.3 · p. 6

3.4. Keeping the map a diffeomorphism: the fold barrier

  • A one-sided Jacobian-determinant barrier activates near zero to discourage folding while allowing positive determinants small enough for volume contraction.Folding would make the map non-invertible and disrupt the correspondence between physical and reference problems.Source: §3.4 · p. 6

3.5. Boundary-preservation constraint

  • Projecting displacement onto the boundary tangent plane removes its normal component, preserving the domain boundary exactly without a penalty parameter.The projector is applied as a mask to the displacement for every parameter value.Source: §3.5 · p. 7

3.6. Training objective on the reference domain

  • The reference-domain objective evaluates boundary data at each boundary point's physical image, allowing points to slide while retaining the correct prescribed data.The reference collocation grid stays fixed while the mapped physical points evolve during joint training.Source: §3.6 · p. 7

4. Joint training of network weights and deformation parameters

The method jointly optimizes solution-network and deformation parameters using Adam, with separate learning-rate schedules and an initial identity-map warm-up. The deformation adds computation for evaluating the map and its derivatives, while its parameter count is small relative to the solution network in the reported two-dimensional settings.

4. Joint training of network weights and deformation parameters

  • Training freezes the deformation at the identity for an initial warm-up, then jointly optimizes network and deformation parameters with Adam and separate learning rates.The optimizer buffers and cosine schedules are restarted when the deformation is unfrozen.Source: §4 · p. 8
  • A single reverse-mode pass obtains gradients for both parameter groups from the reference-domain loss.Automatic differentiation computes the reference-domain fluxes and source terms.Source: §4 · p. 8
  • For the two-dimensional problems, the deformation uses at most 500 parameters versus about 10^5 for the solution network, with added per-batch cost scaling as NkNB.The overhead comes from evaluating the RBF sum and derivatives and assembling det G and G^-1.Source: §4 · p. 8

5. Numerical experiments

The experiments compare FT-PINN with a vanilla PINN under otherwise identical architecture, optimizer, collocation budget, and training horizon. They use viscous formulations and assess accuracy with measures tailored to the distinction between viscous layers and inviscid shocks.

5. Numerical experiments

  • FT-PINN is compared with a vanilla PINN using the same architecture, optimizer, collocation budget, and training horizon.FT-PINN uses jointly trained Gaussian RBF deformation and solution networks; the vanilla model uses a uniform collocation grid.Source: §5 · pp. 8–9
  • Euler cases use artificial viscosity to make pointwise derivatives and the strong-form residual well-defined, producing internal layers of width O(ν) rather than jumps.Burgers cases use physical viscosity, while the Euler model is physically inviscid.Source: §5 · p. 9
  • For Sections 5.2–5.4, accuracy is assessed primarily through feature profiles and full-field contours because pointwise error against inviscid references is floored by regularization.For the decelerating shock, pointwise error is also shown on a logarithmic scale with ν marked.Source: §5 · p. 9
  • Unless stated otherwise, experiments use a five-layer, width-128 FCNN with GELU and random Fourier features, Gaussian RBF displacement, and Adam with initial learning rates of 10^-3.Problem-specific settings are collected in Table 1.Source: §5 · p. 9

Setup.

The experiments test FT-PINN on moving and merging Burgers shocks, Euler waves in space-time, and steady two-dimensional shock reflection. Across these cases, the reported comparisons show FT-PINN recovering shock structures that vanilla PINNs smear or fail to form.

Setup.

  • FT-PINN reproduces the merging-shock reference and tracks both shock trajectories, while the vanilla PINN smears both fronts and misses the post-merge shock.The FT-PINN profiles remain superposed on the reference at t = 0.1, 0.3, 0.6, and 0.9.Source: Setup · p. 10

5.2. Space-time Burgers with a decelerating shock

  • FT-PINN captures the decelerating Burgers shock at its correct location, with error of order ν outside the viscous layer; the vanilla PINN never develops a jump.The vanilla PINN error is O(1) over a wide band around the shock at each reported time.Source: §5.2 · p. 12

5.3. One-dimensional Euler shock tube in space-time

  • FT-PINN approximates the Euler shock-tube rarefaction, contact, and shock, while the vanilla PINN resolves none of the three waves.The map clusters points strongly at the shock, less along the rarefaction, and not at the contact; the contact is nevertheless captured well despite its low residual.Source: §5.3 · pp. 13–14

5.4. Steady two-dimensional Euler regular shock reflection

  • FT-PINN recovers the regular-reflection two-shock pattern on the 98 × 58 grid, whereas the uniform-grid vanilla PINN forms neither shock and the random-point baseline broadens both legs.The FT-PINN places the reflection point near x1,imp = 0.935 and closely approximates the pressure and density jumps.Source: §5.4 · p. 16

6. Conclusion

FT-PINN jointly trains a reference-domain solution network and a deformation map, using the map to concentrate physical collocation points along shocks. Across four shock-dominated problems it resolves the features where a vanilla PINN with the same architecture, budget, and training horizon fails.

6. Conclusion

  • A reference-domain solution network is composed with a jointly trained deformation map, so a uniform reference grid concentrates physical collocation points along features without prior location knowledge or pointwise optimization.The map uses a mixture of radial basis functions; a one-sided Jacobian fold barrier discourages folding, while a tangential projector preserves the boundary exactly and permits boundary sliding.Source: §6 · p. 18
  • Across all four shock-dominated problems, FT-PINN resolved the feature where a vanilla PINN with the same architecture, collocation budget, and training horizon failed.The tests cover merging and decelerating Burgers shocks, a space-time Euler shock tube, and steady 2D Euler regular shock reflection; the current study is limited to rectangular domains and two-dimensional problems.Source: §6 · p. 18

Appendix A. Exact solution of the decelerating-shock Burgers benchmark

The appendix derives an entropy solution and shock path for the decelerating-shock Burgers benchmark. Its characteristic analysis shows that the initial jump generates the shock, while the smooth branch is obtained from the unique characteristic reaching each point left of it.

Appendix A. Exact solution of the decelerating-shock Burgers benchmark

  • The appendix derives the benchmark’s entropy solution and parametric shock path, then verifies that the path satisfies Rankine–Hugoniot identically.Source: Appendix A · p. 20

Appendix A.1. Setup

  • The inviscid Burgers setup uses a quadratic initial ramp on the left and zero state on the right, with zero boundary flux and conserved total mass.The initial ramp rises from 0 at x1 = -1 to 2 at x1 = 0−, then jumps to zero.Source: Appendix A.1 · p. 20

Appendix A.2. Characteristics and the smooth branch

  • Characteristics from the ramp are straight lines carrying constant state and do not intersect; the zero-state right characteristics are vertical, so the jump at x1 = 0 produces a shock immediately.Source: Appendix A.2 · p. 21
  • Left of the shock, each point is reached by a unique ramp characteristic, yielding the closed-form smooth branch; the state is zero at and to the right of the shock.Source: Appendix A.2 · p. 21

Appendix A.3. The shock curve

The shock path is determined by the Rankine–Hugoniot speed and conservation of mass as it absorbs incoming characteristics. The resulting shock decelerates, remains within the spatial domain through t = 1, and remains entropic.

Appendix A.3. The shock curve

  • The Rankine–Hugoniot speed equals half the left state, and the mass balance closes the path; differentiating it recovers the jump-condition speed identically.Source: Appendix A.3 · pp. 21–22
  • Because the right state is zero and the shock speed lies between the right and left characteristic speeds, the shock is Lax-admissible.Source: Appendix A.3 · p. 21
  • The shock absorbs characteristics in decreasing label order, with t(a) = (1 − a^3)/(3a^4) and x_s(a) = (a − 1) + 2a^2t tracing its path.Source: Appendix A.3 · pp. 21–22
  • The shock speed decreases monotonically from 1 at birth to about 0.47 at t = 1, when its position is about 0.636; it stays within [-1, 1].Absorption continues because arriving characteristics travel at twice the instantaneous shock speed; asymptotically, the shock slows algebraically without stopping.Source: Appendix A.3 · p. 22

Appendix B. Exact solution of the regular shock reflection benchmark

The appendix derives the analytical three-state solution for regular shock reflection by applying oblique-shock relations across the incident and reflected legs. The prescribed flow turn determines the shock-angle regime, with detachment bounding regular reflection.

Appendix B. Exact solution of the regular shock reflection benchmark

  • The three-state regular-reflection solution applies oblique-shock relations first across the incident leg and then the reflected leg, with the top-wall slip condition closing the system.The configuration consists of uniform states separated by incident and reflected shocks; the streamline turns across each leg and is wall-parallel in the final state.Source: Appendix B · p. 22, Figure B.15 · p. 23

Appendix B.1. Oblique-shock relations

  • Across an oblique shock, tangential velocity is continuous while normal velocity drops, rotating the velocity vector and deflecting the flow; the normal Mach number drives the normal-shock jumps.Source: Appendix B.1 · p. 22
  • For a prescribed deflection below its maximum, two shock angles are possible and the attached-shock construction uses the weaker root; exceeding the maximum causes detachment and bounds regular reflection.Source: Appendix B.1 · pp. 22–23

Appendix B.2. The incident shock

The incident shock is constructed from the specified freestream and wall deflection, yielding a post-shock state and a leg that reaches the top wall.

Appendix B.2. The incident shock

  • For freestream state W1 = (1, 3.431, 0, 1), M1 = 2.90, and imposed deflection θw = 10°, the weak-root calculation gives M2 = 2.423.Source: Appendix B.2 · p. 23
  • The downstream speed is V2 = 3.184, with velocity components u2 = 3.136 and v2 = 0.553.Source: Appendix B.2 · p. 23
  • The incident leg starts at the inlet split point (0, 0.5) and meets the top wall x2 = 1.Source: Appendix B.2 · p. 23

Appendix B.3. The reflected shock and the three-state field

The reflected shock turns the post-incident flow back to wall-parallel state; the specified 10° turn is below the attachment limit, producing a regular reflection.

Appendix B.3. The reflected shock and the three-state field

  • The 10° wall turn is below the 28.95° detachment value at M2 = 2.423, so an attached weak reflected shock exists and the reflection is regular.Source: Appendix B.3 · p. 24
  • The reflected shock is not specular: its upstream Mach number is M2 = 2.423 rather than the incident shock's M1 = 2.90.Source: Appendix B.3 · p. 24
  • The reflected-shock jump relations give M3 = 2.019 and downstream speed V3 = 2.904.Source: Appendix B.3 · p. 24
  • The reflected-shock turn restores wall-parallel state W3, with u3 = 2.904 and v3 = 0, satisfying the top-wall slip condition exactly.Source: Appendix B.3 · p. 24
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