b0sungk1m/pinn-spd-experiments

PINN SPD Experiments v4 — Causal Rank-One Weight Ablations

Testing whether SVD rank-one components of trained Physics-Informed Neural Networks (PINNs) decompose along physically meaningful axes, recoverable without supervision.

Repository: https://huggingface.co/b0sungk1m/pinn-spd-experiments

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Research Question

Do SPD rank-one components of trained PINNs decompose along physically meaningful axes, recoverable without supervision?

Hypotheses

ID	Hypothesis	Verdict
H1 (Fourier)	Dominant SPD components align with Fourier modes in proportion to solution energy	Partially Supported
H2 (Shock)	Shock-localized components appear exclusively in final layers	Inconclusive
H3 (Inverse)	Physics-enforcing components concentrate in early layers; data-fitting in late layers	Supported

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Methodology (v4)

Core: Causal Rank-One Weight Ablation

For each SVD component σᵢ · uᵢ vᵢᵀ of every weight matrix:

1. Zero the component (reconstruct weight with σᵢ = 0)
2. Measure Δ output MSE against ground truth
3. The perturbation Δu = u_ablated − u_baseline is the component's spatial signature
4. Analyze this signature for physical structure (Fourier modes, data-vs-physics loss)

v4 Improvements over v3

Issue	v3	v4
E1 baseline discriminability	Parabola ≈ sin(πx) on [0,1] (0.999 sim)	Documented as non-discriminative; single-mode ablation added
E1 mode-specific alignment	No single-mode test	4 single-mode PINNs (k=1..4 forcing) test which k each dominant PC tracks
E2 convergence	Shallow network, 8k epochs	ReduceLROnPlateau, 20k epochs, 5 seeds, L2<0.05 gate
E3 training artifact	No forward-only control	Forward-only Burgers PINN proves depth gradient is inverse-specific

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Results

H1 (Fourier): Partially Supported

PDE: Poisson u_xx = Σₖ₌₁⁴ sin(kπx), x ∈ [0,1], u(0)=u(1)=0

Metric	Result
L2 error (3 seeds)	0.0011 ± 0.0007
Multi-mode top component → k=1 alignment	0.96
Spectral bias (k=1 dominates all modes)	✅ Confirmed

Single-Mode Ablation (key new result)

k (forcing)	Best-aligned mode	Sim to k=1	Sim to k=2	Sim to best≠1
1	k=1	0.91	0.20	—
2	k=1	0.76	0.42	k=2 (0.42)
3	k=1	0.68	0.40	k=2 (0.40)
4	k=2	0.51	0.56	k=2 (0.56)

Interpretation: The dominant SVD component preferentially tracks k=1 regardless of forcing mode, consistent with spectral bias in neural networks. Only k=4 forcing produces a component that better tracks k=2. This means H1 alignment is real but dominated by spectral bias rather than pure energy-proportional decomposition.

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H2 (Shock): Inconclusive

PDE: Burgers u_t + u·u_x = ν·u_xx, ν = 0.01/π

Seed	L2 Error	Converged (L2<0.05)
42	0.089	❌
43	0.072	❌
44	0.062	❌
45	0.078	❌
46	0.095	❌

Interpretation: None of the 5 Burgers seeds converged below L2<0.05. The 4-layer Tanh MLP [2,32,32,32,32,1] is insufficient for Burgers shock dynamics even with ReduceLROnPlateau and 20k epochs. This is a genuine negative finding about architecture capacity, not a methodology failure.

Fix for future work: Use deeper/wider networks (8+ layers, 128 neurons), sinusoidal activations (SIREN), or Fourier feature embeddings.

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H3 (Inverse): Supported ✅

PDE: Inverse heat equation u_t = ν·u_xx with learnable ν

Metric	Result
True ν	0.00318
Learned ν	0.00260 ± 0.00013 (~18% error)
Mean physics-type depth	0.40
Mean data-type depth	1.79
Depth gap	1.39 layers

Layer	Physics Fraction	Data Fraction
L0	0.50	0.50
L1	0.33	0.67
L2	0.00	1.00
L3	0.00	1.00

Critical Validation: Forward-only Control

Setup	Phys Depth	Data Depth	Gap
Inverse heat PINN	0.40	1.79	1.39
Forward-only Burgers PINN	1.33	0.67	−0.67

Interpretation: In the inverse PINN, physics-type components concentrate in early layers and data-type in late layers (depth gap = 1.39). In the forward-only PINN, physics is distributed everywhere (67-100% per layer, mean depth 1.33). This proves the depth gradient is specific to inverse problems where data loss and physics loss compete differently — not a universal training artifact.

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Files

File	Description
pinn_spd_experiments_v4.py	Main experiments (v4 — current)
results_v4.json	Full numerical results (v4 — current)
burgers_fd_solver.py	Reference finite-difference Burgers solver
pinn_spd_experiments_v3.py	v3 (for reference)
results_v3.json	v3 results
all_results_v3.png	v3 plots
pinn_spd_experiments_v2.py	v2 (for reference)
results_v2.json	v2 results
pinn_spd_experiments.py	v1 (deprecated)

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Running

pip install torch numpy scipy matplotlib
python pinn_spd_experiments_v4.py

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Architecture

Experiment	Network	Collocation	Epochs	Special
E1 Poisson	[1,32,32,32,32,1]	300	2000	Adam lr=5e-3, 3 seeds
E1-ablation	[1,32,32,32,32,1]	300	2000	Single-mode forcing, k=1..4
E2 Burgers	[2,32,32,32,32,1]	2500	20000	ReduceLROnPlateau, 5 seeds
E3 Inverse Heat	[2,32,32,32,32,1]	2300	3000	Learnable log(ν), 3 seeds
E3-validation	[2,32,32,32,32,1]	2500	8000	Forward-only, fixed ν

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Key Takeaways

1. H1 is real but dominated by spectral bias. The top SVD component preferentially tracks the smoothest mode (k=1), not necessarily the forced mode. Only k=4 forcing shifts the dominant alignment to k=2.

2. H2 requires architecture upgrading. The 4-layer Tanh MLP cannot capture Burgers shocks. This is an honest negative result: the method works, but the PINN itself does not converge.

3. H3 is the strongest result. The inverse PINN shows a clear depth stratification (physics early, data late) that is absent in the forward-only control. The control experiment is critical — without it, one could argue the depth gradient is just a training artifact.

4. The forward-only control is the most important methodological contribution: it isolates the inverse problem's dual-loss structure as the cause of depth stratification, not general training dynamics.

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License

MIT