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Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees
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Proposes a PINN-based solver for functional differential equations by first applying a cylindrical (orthonormal-basis) approximation that turns the FDE into a high-dimensional PDE, with convergence proofs for approximated functional derivatives and solutions.
Problem
Numerically solving functional differential equations (FDEs) is computationally intractable with discretization, and existing cylindrical-approximation solvers are limited to very small basis degrees (m ≤ 6) and oversimplify solutions.
Approach
Expand input functions in an orthonormal basis (cylindrical approximation) to convert an FDE into a high-dimensional PDE over expansion coefficients, prove convergence of approximated functional derivatives and solutions, and solve the resulting PDE mesh-free with a PINN, achieving O(m^r) cost instead of O(m^6).
Key results
- On the functional transport equation, the PINN approach achieves mean relative L1 errors around 1e-3 at degrees 4, 20, and 100 (e.g., 1.27e-3 at degree 4 under linear initial condition).
- On the Burgers-Hopf equation, relative errors of order ~1e-3 are reported with degrees up to 100 (e.g., 2.94e-4 at degree 4 with delta initial condition).
- The method scales to degree 1000 (nonlinear FTE: 8.27e-3 relative error), extending the degree range beyond the m ≤ 6 limit of CP-ALS-based solvers.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase cylindrical approximation degree from 4 to 1000 | FunctionalPINN · FTE, nonlinear initial condition | relative error: $(1.79295 \pm 0.28535) \times 10^{-3}$ at degree 4; $(8.27096 \pm 1.19378) \times 10^{-3}$ at degree 1000 | Hurt |
| Increase cylindrical approximation degree from 4 to 100 | FunctionalPINN · FTE, linear initial condition | relative error: $(1.26820 \pm 0.31421) \times 10^{-3}$ -> $(6.24740 \pm 0.33492) \times 10^{-3}$ | Hurt |
| Increase cylindrical approximation degree from 4 to 100 | FunctionalPINN · BHE, delta initial condition | relative error: $(2.93905 \pm 0.17403) \times 10^{-4}$ -> $(2.41667 \pm 0.25264) \times 10^{-4}$ (degree 100: $(2.41667 \pm 0.25264) \ti | Mixed |
| Add a loss term for the identity W([0], t) = 0 (ablation) | FunctionalPINN on BHE · BHE, functional derivative at θ=0 | relative error: 10^{-1} | Helped |
Surprising findings
- Relative error does not monotonically decrease with degree; errors are comparable or higher at degree 100 and 1000 than at degree 4 in several settings.
Limitations
- Experiments limited to 1+1 spacetime dimensions.
- Experiments limited to first-order functional derivatives (r=1); second order only sketched.
- Covers only abstract evolution equations, not e.g. Schwinger-Dyson or Wetterich equations.
Experimental setup
Datasets: synthetic analytic FDE instances (functional transport equation, Burgers-Hopf equation)
Benchmarks: functional transport equation (linear and nonlinear initial conditions) Burgers-Hopf equation (delta and constant initial conditions)
Metrics: L1 relative error L1 absolute error
Baselines: CP-ALS hierarchical Tucker
Builds on: physics-informed neural networks cylindrical approximation CP-ALS
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