Physics-Informed Functions & SMSF

• Physics-Informed Functions & SMSF are neural network-based approximators that embed PDE constraints to inherently satisfy physical laws and certify stability.
• The framework integrates PINN loss design using PDE residuals, boundary conditions, and fractional score functions to address high-dimensional challenges.
• It provides actionable insights for computing domains of attraction, verifying control Lyapunov functions, and achieving rigorous certification in complex dynamical systems.

Physics-informed functions constitute a class of neural network-based function approximators that, by construction, satisfy or approximate key physical laws expressed as partial differential equations (PDEs). Within this framework, physics-informed neural networks (PINNs) embed the governing equations, such as Lyapunov PDEs, Hamilton–Jacobi–Bellman (HJB) equations, or Fokker–Planck–Lévy (FPL) equations, directly into the loss function. Satisfiability modulo theories (SMT) formal verification (here referred to as SMSF) can be used in conjunction with these neural approximators to attain mathematically certified guarantees for stability, controllability, or other safety- or physics-critical properties. Recent research leverages these tools for stability verification, computation of domains of attraction (ROA), high-dimensional fractional PDEs, and optimal control, fundamentally advancing both computational methodology and verifiable certification in nonlinear dynamics, stochastic processes, and control.

1. Physics-Informed Neural Network Lyapunov and Zubov Functions

Physics-informed neural networks for Lyapunov and Zubov functions are based on explicit PDE characterizations of the requirements for Lyapunov stability and for the exact domain (ROA) to which solutions are attracted. For a system $\dot{x}=f(x)$ with $f\colon\mathbb{R}^n\to\mathbb{R}^n$ and stable equilibrium at $x=0$, the Lyapunov condition can be recast as a first-order linear PDE:

$DV(x)\cdot f(x) + \omega(x) = 0,$

with positive-definite $\omega$ (the dissipation rate). The unique continuous viscosity solution $V$ is finite only on the ROA and uniquely characterizes it. Zubov’s method extends this by constructing a nonlinear PDE whose unique solution $W$ (under a suitable monotone transformation $\beta$) has its 1-sublevel set exactly equal to the domain of attraction; i.e.,

$DW(x)\cdot f(x) + \omega(x)\psi(W(x))(1-W(x)) = 0,$

with $W(0)=0$ and $f\colon\mathbb{R}^n\to\mathbb{R}^n$0 on the boundary. These results guarantee existence, uniqueness, and strong error bounds for the PDE approach (Liu et al., 2023).

2. PINN Design, Loss Construction, and Convergence

PINNs are constructed as feedforward neural networks $f\colon\mathbb{R}^n\to\mathbb{R}^n$1 with standard architectural choices (hidden layers, activation functions). Training data consists of interior collocation points for the PDE residual, boundary points, and (optionally) numerical trajectories for fitting known values. The training loss combines:

• PDE residual loss (physics fidelity),
• Boundary condition residual,
• Data fitting (optional).

Formally, the loss for the Zubov PDE is

$f\colon\mathbb{R}^n\to\mathbb{R}^n$2

where $f\colon\mathbb{R}^n\to\mathbb{R}^n$3 are PDE residuals, $f\colon\mathbb{R}^n\to\mathbb{R}^n$4 are boundary residuals, $f\colon\mathbb{R}^n\to\mathbb{R}^n$5 are data fitting errors (Liu et al., 2023, Liu et al., 2024). Automatic differentiation is used to compute all required network gradients.

Convergence guarantees are derived from viscosity solution theory: the PINN output converges uniformly to the true $f\colon\mathbb{R}^n\to\mathbb{R}^n$6 if residual errors and boundary mismatches vanish and the network class is sufficiently expressive.

3. Formal Verification via SMT (SMSF)

Neural Lyapunov functions—even physics-informed—require post-training verification to guarantee formal satisfaction of stability or optimality properties.

• For local exponential stability, certificates are derived by encoding conditions such as $f\colon\mathbb{R}^n\to\mathbb{R}^n$7 as SMT queries, where $f\colon\mathbb{R}^n\to\mathbb{R}^n$8 is a quadratic Lyapunov function.
• For global ROA or null-controllability sets, more general sublevel sets of the neural function (e.g., $f\colon\mathbb{R}^n\to\mathbb{R}^n$9) are verified by asserting, for all $x=0$0 in a prescribed domain or under side conditions, that the required decrease condition (e.g., $x=0$1) and set containment ($x=0$2) hold (Liu et al., 2023, Liu et al., 2024).

Tools such as dReal and Z3 are employed to automatically search for, or rule out, counterexamples, providing machine-checkable certificates that the neural network truly functions as a Lyapunov or control Lyapunov function.

4. Physics-Informed Neural Control Lyapunov Functions (CLFs) and Zubov-HJB Transform

For controlled nonlinear systems, PINNs can be trained to solve a transformed HJB equation that, after Zubov-type transformation, characterizes both the optimal value function and the null-controllability set. The transformed equation has the form:

$x=0$3

where $x=0$4 is the optimal feedback determined by Pontryagin’s maximum principle (PMP) and $x=0$5 is the stage cost. Trajectory-wise data computed via the PMP two-point boundary value problem directly inform the loss, enforcing pointwise correctness in addition to PDE satisfaction (Liu et al., 2024). The certified neural CLF thus directly approximates the maximal (null-)controllability set, and its level sets are verifiable by SMT.

5. Fractional Score-Based PINNs and the Curse of Dimensionality

For high-dimensional stochastic systems described by Fokker–Planck–Lévy equations,

$x=0$6

standard PINNs fail due to the curse of dimensionality (CoD) and severe numerical underflow. Score-fPINN introduces a fractional score function

$x=0$7

enabling elimination of the nonlocal fractional Laplacian and reducing the problem to a local, second-order PDE for $x=0$8. This transformation ensures that the critical function $x=0$9 remains $DV(x)\cdot f(x) + \omega(x) = 0,$0 as dimension $DV(x)\cdot f(x) + \omega(x) = 0,$1 increases, allowing stable training and accuracy even for $DV(x)\cdot f(x) + \omega(x) = 0,$2. Two main strategies exist:

• Fractional Score Matching (FSM): efficient if conditional densities are known, but limited in applicability.
• Score-fPINN: more general, but requires computationally intensive automatic differentiation (Hu et al., 2024).

Empirical results across SDE benchmarks confirm sublinear scaling in $DV(x)\cdot f(x) + \omega(x) = 0,$3 and robust accuracy even in heavy-tailed regimes.

6. Comparative Analysis and Methodological Impact

Physics-informed, neural PDE-based approaches surpass traditional sum-of-squares (SOS) and rational function methods in several aspects:

• Expressivity: Deep networks can model complex, high-dimensional, or non-polynomial ROA shapes inaccessible to low-degree SOS polynomials.
• Scalability: PINNs, particularly when coupled with fractional scoring, remain tractable in dimensions where SOS/SDP methods become infeasible (e.g., $DV(x)\cdot f(x) + \omega(x) = 0,$4–$DV(x)\cdot f(x) + \omega(x) = 0,$5 or FPL equations up to $DV(x)\cdot f(x) + \omega(x) = 0,$6).
• Verification: Integration with SMT solvers provides machine-checked guarantees, essential for safety-critical applications.
• Empirical Tightness: PINN-based estimates for ROA or controllability sets routinely approach or surpass the tightest known polynomial bounds, with higher coverage evident in empirical benchmarks (Liu et al., 2023, Liu et al., 2024).

Limitations include non-convex optimization (no global optimum guarantee), hyperparameter sensitivity, and computational cost of SMT in very high dimensions.

7. Applications and Extensions

Key applications include:

• Nonlinear stability region computation and certification,
• Feedback stabilization via neural control Lyapunov functions,
• Solving high-dimensional stochastic PDEs in finance, physics, and biology,
• Data-driven inverse identification of system parameters.

The methodologies developed—physics-informed loss construction, fractional scores, Zubov transformations, and formal verification—are extensible to inverse problems, general Lévy processes, hybrid PINN–domain decomposition methods, and other settings demanding certifiable neural approximations of physical laws.

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References

• "Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification" (Liu et al., 2023)
• "Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations" (Hu et al., 2024)
• "Formally Verified Physics-Informed Neural Control Lyapunov Functions" (Liu et al., 2024)