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Physics and Lie symmetry informed Gaussian processes
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Introduces physics and Lie symmetry informed Gaussian processes (PSGP), which constrain a physics-informed GP with a Lie symmetry via virtual invariant-surface-condition observations or kernel design in canonical coordinates, and compares against PIGP, PINN and PSNN on Fisher, heat and wave PDEs.
Problem
Physics-informed machine learning for PDEs is data-hungry, and known Lie symmetries of PDEs are rarely exploited in PIML frameworks, especially in GP models.
Approach
Incorporate a Lie symmetry into a physics-informed GP as an induced invariant surface condition (ISC) imposed through virtual observations in a joint variational GP model, or explicitly by defining the kernel on canonical coordinates so the ISC and PDE hold exactly.
Key results
- On the Fisher-like equation with 16 virtual observations, PSGP achieved test MAE 360e-5 versus 1120e-5 for PIGP and 37600e-5 for PINN.
- On the wave equation with 16 observations, PSGP achieved MAE 14.9e-2 versus 66.7e-2 for PIGP and 24.5e-2 for PINN, with PSGP best for Nu <= 128.
- For the heat inverse problem, PSGP outperformed PIGP in parameter error, MAE and CRPS, and recovered the solution almost perfectly with Nu=10 observations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Lie-symmetry ISC observations to the PIGP | PIGP · Fisher-like equation | MAE: 1120 -> 360; 230 -> 44.9; 25.6 -> 10.3; 5.1 -> 3.3 | Helped |
| Add Lie-symmetry ISC observations to the PIGP | PIGP · Heat equation inverse problem | parameter inference accuracy | Helped |
| Add Lie-symmetry ISC observations to the PIGP | PIGP · Heat equation inverse problem | MAE | Helped |
| Add Lie-symmetry ISC observations to the PIGP | PIGP · Heat equation inverse problem | CRPS | Helped |
| Use the canonical-coordinate kernel to explicitly enforce the ISC and PDE | PIGP · Wave equation | MAE: 66.7 -> 14.9; 66.9 -> 4.4; 66.9 -> 4.0; 51.3 -> 3.2; 3.1 -> 1.5 | Helped |
| Use a periodic kernel instead of the canonical-coordinate PSGP kernel (ablation) | PSGP · Wave equation | prediction error | Helped |
| Increase the number of Fisher-equation collocation points | PSGP · Fisher-like equation | MAE: 360; 44.9; 10.3; 3.3 | Helped |
Surprising findings
- For the wave equation, using more collocation points made PIGP worse, as the trivial solution attractor covered the parameter space.
- The PIGP learned the trivial solution near zero for sparse wave-equation data with Nu <= 128.
Limitations
- GP cubic complexity restricts the method to low to medium-sized datasets.
- Lie symmetry is assumed known a priori and restricted to classical symmetries.
Experimental setup
Datasets: synthetic Fisher-type equation solutions synthetic heat equation data synthetic wave equation data
Benchmarks: Fisher-like equation heat equation inverse problem wave equation forward problem
Metrics: MAE absolute parameter error |theta - theta_hat| CRPS
Baselines: PIGP PINN PSNN
Builds on: physics-informed Gaussian processes Long et al. 2022 nonlinear PIGP variational framework symmetry-enhanced PINN Raissi et al. 2017 PIGP
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