Inverse PINN: Parameter Recovery & Inference

• Inverse PINNs are methods that embed governing physics into neural network loss functions to accurately infer unknown parameters in PDE systems.
• They employ diverse architectures including parametric, function-valued, and hybrid models to tackle complex inverse problems across various applications.
• Advanced optimization, adaptive loss weighting, and uncertainty quantification strategies enable robust recovery even under sparse and noisy measurement conditions.

Inverse Physics-Informed Neural Networks (inverse PINNs) refer to a class of methodologies for inferring unknown parameters, fields, or source terms in differential equation-governed systems by embedding physical knowledge into the loss function of neural network surrogates. Unlike forward PINNs, which solve for state variables given equations and parameters, inverse PINNs leverage physical laws and sparse or noisy observations to estimate unknown constitutive parameters, source terms, latent fields, or structural design variables. This class of methods is foundational for scientific machine learning, enabling data-efficient and physically consistent solutions to inverse problems in systems modeled by ordinary and partial differential equations.

1. Mathematical Formulation and Core Principles

Given a parameterized physical system defined by a (possibly nonlinear or nonlocal) differential operator

$$\mathcal{D}[u(x,t);\theta] = 0,\quad x\in\Omega,\, t\in\mathcal{T}$$

subject to initial/boundary conditions and possibly with unknown parameters $\theta$ or unknown spatially varying fields, the inverse PINN seeks to reconstruct both $u$ and $\theta$ (or fields such as $c(x), \mu(x)$, or $f(x)$) from partial and potentially noisy measurements. The typical objective is to minimize a composite loss function of the form

$$\mathcal{L} = w_{\text{data}}\, \mathcal{L}_{\text{data}} + w_{\text{pde}}\, \mathcal{L}_{\text{pde}} + w_{\text{bc}}\, \mathcal{L}_{\text{bc}} + \dots$$

where $\mathcal{L}_{\text{data}}$ matches network outputs to observed data, $\mathcal{L}_{\text{pde}}$ enforces the residual of the governing PDE (via automatic or numerical differentiation), and $\mathcal{L}_{\text{bc}}$ imposes boundary/initial conditions. In the inverse configuration, unknown $\theta$ (scalars, vectors, or fields) are treated as trainable parameters or network outputs.

Strategies for inverse PINNs encompass parametric identification (physical constants, material parameters), nonparametric regression (spatially varying coefficients or sources), and field recovery in ill-posed scenarios such as Electrical Impedance Tomography or elastography (Yin et al., 2022, Xuanxuan et al., 10 Dec 2024).

2. Inverse PINN Architectures and Implementation Variants

Inverse PINNs are realized through diverse architectures, reflecting problem structure:

• Parametric inverse PINNs: Unknown constants (e.g., diffusion coefficients, Lamé parameters, transport coefficients) are appended to network parameters and optimized alongside weights (Kag et al., 2023, Lu et al., 2023, Almanstötter et al., 7 Apr 2025, Wu et al., 12 Nov 2024).
• Function-valued inverse PINNs: Unknown spatially varying fields (e.g., conductivity $\sigma(x)$, modulus $\mu(x)$, source $f(x)$, variable coefficients $c(t)$) are modeled by auxiliary networks, either as separate “branches” (trunk-branch structures) or as multivariate outputs (Yin et al., 2022, Xuanxuan et al., 10 Dec 2024, Wi et al., 3 Nov 2024, Miao et al., 2023, Xing et al., 21 Jan 2025).
• Nonlocal and constrained problems: Inverse PINN frameworks are adapted to nonlocal and $PT$-symmetric PDEs by recasting integral/nonlocal terms as coupled local variables and embedding symmetry constraints in the loss (Peng et al., 2023).
• Bayesian and uncertainty-aware PINNs: Ensemble (Jiang et al., 2022), Bayesian (Sun et al., 21 Jun 2024), and randomized approaches (Zong et al., 5 Jul 2024) yield pointwise uncertainty quantification in the inferred fields or parameters.
• Hybrid frameworks: Combinations of PINN surrogates with CNNs or mode-matching solvers enable full-inverse recovery from boundary data, as in EIT (Xuanxuan et al., 10 Dec 2024) or frequency-selective surface design (Liu et al., 8 Jan 2024).

A distinguishing feature in these methods is the explicit or implicit joint parameterization of both state and unknowns, with automatic differentiation facilitating gradient-based optimization of all unknowns within a unified computational graph.

3. Loss Construction, Optimization Strategies, and Constraints

The central methodological innovation in inverse PINNs is the loss function construction, reflecting both physical fidelity and data consistency:

• Physics-informed loss: For each collocation point, the residual of the governing PDE is computed by differentiating the network prediction; this enforces the governing equations even where measurements are absent.
• Data misfit: At observation points (usually sparse and noisy), the difference between the predicted and measured quantities forms a standard regression loss.
• Boundary/initial losses: Enforced either as hard or soft constraints in the total loss.
• Auxiliary constraints: For inverse design, symmetry, or feasibility (e.g., non-negativity of physical parameters, as in conductivity recovery (Xuanxuan et al., 10 Dec 2024)).
• Sampler/weighting strategies: Adaptive, epoch-dependent weighting of loss components is common, to facilitate balanced convergence and avoid domination by poorly scaled losses (Berardi et al., 15 Jul 2024, Almanstötter et al., 7 Apr 2025).

Optimization is performed via standard first-order methods (Adam, Adan) often followed by quasi-Newton (L-BFGS) refinement. In multi-objective and multi-constraint scenarios, methods such as the Modified Differential Method of Multipliers (MDMM) (Almanstötter et al., 7 Apr 2025) or NSGA-II (Lu et al., 2023) are employed to discover Pareto-optimal tradeoffs or to enforce constraints exactly.

In challenging settings, dynamic reweighting, gradient scaling, and variable scheduling are crucial to enable stable identification, especially when there are competing objectives or multiple unknowns with heterogeneous sensitivities.

4. Representative Applications and Quantitative Performance

Inverse PINNs have demonstrated efficacy across a range of scientific domains:

• Identification of variable coefficients: VC-PINN achieves L₂ relative errors $10^{-4}$–$10^{-2}$ in recovering nontrivial time-varying coefficients in nonlinear PDEs, robust to noise and convexity challenges (Miao et al., 2023).
• Dynamic material identification: In dynamic elasticity, PINNs recover Lamé parameters to within 2–3% error using sparse boundary data in 2D and 3D, reducing parameter-paper costs by orders of magnitude vs. repeated FEM runs (Kag et al., 2023).
• Full-field elastography and EIT: Simultaneous inference of full modulus fields, contact pressures, and non-smooth conductivities, with relative errors below 2% and robust uncertainty quantification, is achieved in SWENet, Neural Inverse Source Problems, and CPFI-EIT (Yin et al., 2022, Wi et al., 3 Nov 2024, Xuanxuan et al., 10 Dec 2024).
• Nonlocal inverse PDEs: PTS-PINN solves inverse problems in PT-symmetric nonlocal PDEs by re-expressing nonlocal terms as local variables, enabling accurate parameter recovery and reconstructing large-scale nonlinear coherent structures with errors below 0.1% under low noise (Peng et al., 2023).
• Parameter identifiability under noise: PINNverse (with a constrained MDMM approach) demonstrates up to 370× reduction in parameter error and 88× reduction in physics violation compared to unconstrained PINN approaches, maintaining robustness under up to 30% data noise and poor initial guesses (Almanstötter et al., 7 Apr 2025).
• Scalable field inversion with UQ: E-PINN ensemble methods and rPINN randomization facilitate pointwise credible intervals and adaptive sampling, surpassing MC-dropout and deep-ensemble baselines in both accuracy and calibration (Jiang et al., 2022, Zong et al., 5 Jul 2024).

Quantitative performance is frequently assessed by parameter error, field L₂ norm error, and statistical fidelity of Bayesian/posterior samples. Pareto-front exploration and constraint satisfaction are critical metrics when weights/tradeoffs are not prescribed a priori.

5. Challenges, Limitations, and Theoretical Foundations

Inverse PINNs face key challenges:

• Loss landscape complexity and convergence: Non-convexity and conflicting objectives (data fit vs. physics residuals) can trap optimizers in local minima; the need to explore the full Pareto front motivates adoption of constrained optimization (MDMM) and multi-objective algorithms (NSGA-II) (Lu et al., 2023, Almanstötter et al., 7 Apr 2025).
• Noise sensitivity and regularization: Noisy observations degrade accuracy but can be mitigated by physics-informed regularization, adversarial training, and ensemble- or Bayesian-based uncertainty treatments (Jiang et al., 2022, Zong et al., 5 Jul 2024, Sun et al., 21 Jun 2024).
• Scaling and hyperparameter selection: Balancing data and physics losses, especially in high-dimensional or multiscale settings, is critical; dynamic (epoch-wise) weighting helps prevent the dominance of any loss component (Berardi et al., 15 Jul 2024, Almanstötter et al., 7 Apr 2025).
• Ill-posedness in limited-data regimes: Fundamental ill-posedness in PDE parameter identification is ameliorated by strong physical inductive bias and physics-based priors in Bayesian PINNs, with convergence rates characterized theoretically for linear-parameter PDEs (Sun et al., 21 Jun 2024).
• Computational cost: Nontrivial training times (O($10^5$–$10^6$ epochs) for full-field problems are typical; two-stage and hybrid methods (e.g., CNN-PINN) as in EIT (Xuanxuan et al., 10 Dec 2024) reduce overall complexity.
• Generalization to nonlocal/multiphysics scenarios: Extensions to nonlocal, non-smooth, or hybrid physics require careful reformulation of PDE residuals, as in the introduction of "mirror fields" for PT symmetry (Peng et al., 2023), or discrete derivative operators for highly irregular fields (Xuanxuan et al., 10 Dec 2024).

Theoretical results establish that, for linear-parameter PDEs, Bayesian PINN estimators recover solutions and parameters at minimax-optimal rates, with additional convergence penalties for higher-order parameter dependence (Sun et al., 21 Jun 2024).

6. Extensions, Impact, and Future Directions

Inverse PINN methodology is now applied across physics, engineering, computational biology, materials science, and medical/industrial imaging. Innovative architectures—such as trunk-branch splits for global/local feature learning (Xing et al., 21 Jan 2025), multiscale and small-velocity amplification embeddings for highly multiscale flows (Wu et al., 12 Nov 2024), and simulation-driven or hybrid frameworks (Besnard et al., 2023, Xuanxuan et al., 10 Dec 2024)—continue to expand the domain of applicability.

Prospective directions include:

• Robust, adaptive weighting schemes for loss components to support automated, problem-agnostic tuning (Berardi et al., 15 Jul 2024, Almanstötter et al., 7 Apr 2025).
• Nonparametric inference of spatially heterogeneous coefficients using high-capacity networks with uncertainty quantification (Yin et al., 2022, Zong et al., 5 Jul 2024).
• Multi-physics and multi-scale couplings in inverse settings (e.g., multi-frequency or time-domain EIT, poromechanics, turbulence) (Xuanxuan et al., 10 Dec 2024, Wu et al., 12 Nov 2024).
• Integration with experimental design and active control, leveraging uncertainty estimates for targeted measurement or adaptive system identification (Jiang et al., 2022).
• Theoretical advances in convergence, identifiability, and sample complexity in high-dimensional, nonlinear, and ill-posed inverse PDEs (Sun et al., 21 Jun 2024, Almanstötter et al., 7 Apr 2025).

Inverse PINNs constitute a rapidly converging field, grounding machine learning-based inference in fundamental physics while offering strong data efficiency, extensibility, and uncertainty quantification for real-world, data-limited inverse problems.