Symplectic PINNs for stable deep physics-informed learning

Parth Sinha & Shine Gupta · 2026 · Work in progress

Can enforcing symplecticity in a neural network's hidden state prevent gradient collapse at depth? This site documents the experimental evidence for and against that hypothesis.

A separable Hamiltonian hidden state, evolved through Störmer-Verlet leapfrog layers, preserves the Jacobian singular-value spectrum at every depth, keeping deep physics-informed networks trainable where standard MLPs fail catastrophically.

Key results

50 layers tested SymplecticPINN keeps usable gradient signal at depth 50.

6 orders better than MLP Depth-50 heat-equation loss stays near 1e-7 instead of 1e-1.

SV >= 0.25 minimum signal Every phase-space direction remains trainable in the corrected model.

Architecture

The model encodes PDE input coordinates into a hidden phase space, evolves them through N leapfrog layers using a separable Hamiltonian H(q, p) = T(p) + V(q), then decodes to the solution. Because the Störmer-Verlet update is exactly symplectic, det(J) = 1 holds at every layer, so Jacobian singular values cannot collapse.

Project (x, t) → (q₀, p₀) ∈ ℝ^d × ℝ^d Input coordinates into hidden phase space

Evolve ∂T/∂p and ∂V/∂q Volume-preserving leapfrog, N layers

Decode (q_N, p_N) → u(x, t) Phase coordinates to PDE solution

Live visualization

Network depth

Animation speed

1.0×

Playback

Standard PINN

det(J) = 0.2671 degraded

Symplectic Flow PINN det(J) = 1.0000

Experiments

Baseline MLP vs Symplectic PINN Controlled comparison on the 1D heat equation under matched training conditions. Depth Stress Test Jacobian singular-value spectrum at depths 4, 10, 25, and 50. Primary evidence for deep-scaling. Gradient Alignment Test Cosine similarity between PDE residual and boundary gradients throughout training. Gradient Alignment Depth Test Extends the alignment diagnostic across depths. At depth 50 the MLP gradient collapses to numerical zero.

Status

Supported claims

SymplecticPINN converges faster and trains more stably than a matched MLP on the 1D heat equation. verified
The corrected separable Hamiltonian formulation is mathematically symplectic and empirically volume-preserving. verified
Depth-50 experiments show the MLP breaks while SymplecticPINN preserves usable gradient flow. verified
ResidualMLP remains a strong empirical baseline, but its Jacobian spectrum lacks the same structural guarantee. verified

Open caveats

Gradient-alignment depth-test plots should be confirmed as running from the final committed architecture. pending
Hard PDE benchmarks such as Helmholtz and Kuramoto-Sivashinsky are not yet implemented. open
Pareto efficiency, memory, and accuracy comparisons are not yet measured. open
Some depth-stress plots should be regenerated from the final committed architecture for paper submission. pending

Citation

If you reference this work before it is formally published:

BibTeX

@misc{sinha-gupta2026symplectic,
  author = {Sinha, Parth and Gupta, Shine},
  title  = {Symplectic {PINN}s for Stable Deep Physics-Informed Learning},
  year   = {2026},
  note   = {Work in progress}
}