Inverse Physics-Informed Neural Networks

• IPINNs are defined as PINN-based formulations for inverse problems, merging observational data with residuals of governing equations to infer unknown physical quantities.
• They use composite loss functions that couple data mismatch with physics residuals, enforcing boundary, initial, and equation consistency through adaptive weighting.
• Applications span parameter identification in porous media and quantum systems to inverse design of metasurfaces, addressing ill-posed inverse problems robustly.

Searching arXiv for the papers on arXiv to ground the article in current literature. Inverse Physics-Informed Neural Networks (IPINNs) are PINN-based formulations for inverse problems in which unknown quantities—such as coefficients, source terms, kernels, geometries, Hamiltonian parameters, or latent physical fields—are treated as trainable variables and inferred by combining observational data with residuals of governing equations, boundary conditions, and initial conditions. Across the recent literature, IPINNs are used both for parameter identification and for function-valued reconstruction, and in several cases they are also used for design, where the neural network parameterizes a candidate object and optimization is driven by physical feasibility and target response rather than a conventional labeled dataset (Liu et al., 2024).

1. Definition and conceptual scope

Within the cited literature, IPINNs are not restricted to a single mathematical setting. They appear in PDE-constrained parameter identification, inverse source recovery, material identification, quantum Hamiltonian learning, inverse design of metasurfaces, and inverse problems with interval, fuzzy, or Bayesian uncertainty descriptions. A recurring formulation is that the network does not merely approximate a forward state variable; it also exposes unknown physical parameters or hidden functions as trainable objects, so that the optimization jointly updates neural-network weights and the unknown physical quantities.

Several papers make this interpretation explicit. For transport models in porous materials, “once a suitable PINN is established to solve the forward problem, the transport parameters are added as trainable parameters” (Berardi et al., 2024). For the heat equation and related didactic models, inverse PINNs “include both neural network parameters $(\theta)$ and the model parameter(s) $D$” among the trainables (Gupta et al., 19 Jul 2025). For radiative transfer, two neural networks are jointly learned: one for the radiative intensity $u$ and one for the absorption coefficient $k$ (Mishra et al., 2020). For inverse source problems, the network learns a mapping

$$\Psi_\theta : \mathbf{x} \mapsto (\Phi_\theta(\mathbf{x}), f_\theta(\mathbf{x})),$$

so that both the hidden state and the unknown source are reconstructed from partial and noisy observations (Wi et al., 2024).

The scope also extends beyond classical coefficient recovery. In the peridynamic setting, the unknown is a nonlocal kernel $C(\cdot)$, learned together with the displacement field $\theta(x,t)$ (Difonzo et al., 2023). In inverse design of frequency selective surfaces, the network output represents diaphragm geometry through binary functions $g_i(x,y)$, and optimization is driven by mode-matching residuals and S-parameter targets rather than by a geometry–response dataset (Liu et al., 2024). This suggests that, in the current arXiv usage, “IPINN” functions as an umbrella term for inverse scientific machine learning with explicit physics-embedded optimization, rather than only for classical coefficient identification.

2. Mathematical formulation

The mathematical core of IPINNs is a composite objective that couples data consistency with equation consistency. The specific decomposition varies by application, but the same structural pattern recurs. In Hamiltonian learning, the total objective is written as

$$\mathcal{L}_\text{total}(w, \bm{\theta}|D) = \mathcal{L}_\text{data}(w, \bm{\theta}|D) + \lambda_1 \mathcal{L}_\text{physics}(w, \bm{\theta}) + \lambda_2 \mathcal{L}_\text{initial}(w, \bm{\theta}|D),$$

where the Schrödinger equation is enforced at sampled physical constraint points and initial states are explicitly penalized (Liu et al., 12 Jun 2025). In inverse radiative transfer, the loss likewise combines a data mismatch term, boundary and initial residuals, and an interior residual obtained by inserting neural-network outputs into the radiative transfer equation (Mishra et al., 2020). In dynamic linear elasticity, inverse material identification uses

$$\mathcal{L}_{\text{tot}} = \lambda_1 L_{\text{data}} + \lambda_2 L_{\text{eqn}},$$

with the Lamé parameters $D$0 and $D$1 treated as trainable variables (Kag et al., 2023).

For many inverse PDE problems, the generic pattern is

$D$2

or an elaboration of it with additional boundary and initial terms. The heat-equation formulation uses

$D$3

with $D$4 optimized jointly with the network (Gupta et al., 19 Jul 2025). For beam systems, the inverse PINN loss is described as a sum of the PDE residual and data mismatch terms, enabling estimation of unknown model parameters and external forces from sparse or noisy observations (Kapoor et al., 2023).

A notable variant appears in inverse design. In the frequency selective surface problem, the unknown geometry is encoded through

$D$5

and the PINN loss is the squared $D$6-norm of the residual of the mode matching equation,

$D$7

augmented by electromagnetic response targets such as prescribed $D$8-parameters (Liu et al., 2024). This formulation illustrates that IPINNs need not infer only coefficients in a differential operator; they can optimize a structural representation when the governing physics can be differentiated through.

3. Architectural patterns and training strategies

The network architectures used in IPINNs are highly problem-dependent, but several design patterns recur. Fully connected feed-forward networks are the most common base architecture. The transport-model study uses a fully connected MLP whose trainables include both network weights and physical parameters such as diffusion, advection velocity, or reaction constants (Berardi et al., 2024). The FSS inverse-design study uses an FCNN with three hidden layers, each with 32 neurons, and the network output directly represents the diaphragm shape $D$9 (Liu et al., 2024). Dynamic elasticity employs 4–5 hidden layers with 128–256 neurons and $u$0 activation, with non-dimensionalized inputs and weighted loss terms (Kag et al., 2023).

Other papers depart from the vanilla architecture to encode stronger inductive structure. Hamiltonian learning uses Neural Network Quantum States, $u$1, as a differentiable representation of a time-dependent quantum state over the computational basis (Liu et al., 12 Jun 2025). In inverse source problems, a modified MLP with a gated “green box” structure and split output heads for state and source was reported to be crucial for inverse-source accuracy (Wi et al., 2024). For peridynamic kernel identification, a serial two-subnetwork architecture is used: an RBF-based kernel network for $u$2, followed by a solution network for $u$3; the use of RBF activations is described as necessary to obtain meaningful solutions consistent with symmetry, smoothness, and locality expectations (Difonzo et al., 2023). For inverse problems in unbounded domains, two independent networks are trained jointly, one for the solution $u$4 and one for the unknown coefficient $u$5, rather than a single network with two outputs (Pérez-Bernal et al., 12 Dec 2025).

Training strategies are equally central. Adam is repeatedly used, sometimes followed by L-BFGS for fine-tuning (Liu et al., 2024, Wickramasinghe et al., 16 Sep 2025). Automatic differentiation is the standard mechanism for computing temporal and spatial derivatives and, in inverse settings, derivatives with respect to physical parameters (Liu et al., 12 Jun 2025, Berardi et al., 2024). Several papers emphasize that loss balancing is not a secondary implementation detail but a determinant of identifiability and convergence. The porous-transport study states that, for convergence to the correct inverse solution, “data misfit, initial conditions, boundary conditions and residual of the transport equation need to be weighted adaptively as a function of the training iteration (epoch),” and that gradients of trainable parameters are scaled at each epoch accordingly (Berardi et al., 2024). The inverse-Dirichlet weighting paper characterizes failures caused by scale-imbalanced multi-objective training and proposes variance-based dynamic weighting to prevent vanishing task-specific gradients and catastrophic forgetting in sequential inverse modeling (Maddu et al., 2021). PBPK-iPINN likewise reports that convergence depends on appropriate weighting of data, initial-condition, and residual losses, as well as careful tuning of layers, neurons, activation functions, learning rate, optimizer, and collocation points (Wickramasinghe et al., 16 Sep 2025).

4. Representative application domains

The application space of IPINNs is broad and technically heterogeneous.

Domain	Inverse target	Representative paper
Electromagnetic metasurfaces	Diaphragm geometry / design	(Liu et al., 2024)
Quantum systems	Hamiltonian parameters	(Liu et al., 12 Jun 2025)
Radiative transfer	Absorption coefficient and intensity	(Mishra et al., 2020)
Transport in porous media	Diffusion, velocity, reaction parameters	(Berardi et al., 2024)
Structural mechanics	Beam parameters, forces, Lamé parameters	(Kapoor et al., 2023, Kag et al., 2023)
Robotics inverse source problems	Source function and full state	(Wi et al., 2024)
Nonlocal wave equations	Peridynamic kernel	(Difonzo et al., 2023)
Pharmacokinetics	Drug-specific or patient-specific PBPK parameters	(Wickramasinghe et al., 16 Sep 2025)
Thermal characterization	Thermal diffusivity	(Zhu et al., 13 Sep 2025)

In quantum characterization, “iPINN-HL” is benchmarked on one-dimensional spin chains, cross-resonance gate calibration, crosstalk identification, and real-time compensation to parameter drift; the method integrates observational data, Schrödinger residuals, and initialization losses, and is reported to approach the Heisenberg limit in certain settings while outperforming a purely data-driven baseline in accuracy and resource efficiency (Liu et al., 12 Jun 2025). In radiative transfer, IPINNs recover unknown coefficients such as $u$6 and reconstruct the radiative intensity $u$7, with theoretical generalization error estimates and a three-dimensional monochromatic steady-state inverse experiment (Mishra et al., 2020).

In mechanics and robotics, inverse PINNs are used to recover applied forces, material parameters, source functions, hidden state trajectories, and contact-related quantities from sparse, noisy, and partial observations. The beam-systems study reports robust inverse recovery of unknown dimensionless parameters and external forces in Euler–Bernoulli and Timoshenko beam systems, including double-beam configurations and noisy data (Kapoor et al., 2023). The robotics-oriented inverse source framework is validated on problems involving up to 4th order PDEs, Signorini and Dirichlet constraints, and regression losses including Chamfer distance and $u$8 norm (Wi et al., 2024).

Inverse design constitutes a distinct but closely related branch. In the FSS study, the network is trained without a conventional dataset, using field solutions from a mode matching method and a physically informed loss to produce diaphragm shapes satisfying single-frequency transmission targets; the reported full-wave simulation exhibits a transmission null at $u$9 GHz relative to a $k$0 GHz design goal, with a relative error of $k$1 (Liu et al., 2024). This suggests that IPINNs can be interpreted not only as solvers of inverse identification problems but also as differentiable physics-constrained optimizers over structural design spaces.

5. Bayesian, simulation-driven, and uncertainty-aware variants

A substantial branch of the literature extends deterministic IPINNs toward probabilistic inference. In “Bayesian Physics-Informed Neural Networks for Inverse Problems (BPINN-IP),” the forward model is cast as

$k$2

with priors placed on unknowns and network parameters, and the standard PINN objective is recovered as a Maximum A Posteriori special case (Mohammad-Djafari et al., 2 Dec 2025). A closely related framework for linear inverse problems derives posterior laws for both unknown variables and neural-network parameters in supervised and unsupervised settings and emphasizes implementation challenges in real applications (Mohammad-Djafari, 18 Feb 2025). These formulations make posterior uncertainty an explicit output rather than an afterthought.

Another probabilistic direction is the physics-informed invertible neural network framework. PI-INN combines an invertible neural network with a neural basis network, learns expansion coefficients and latent Gaussian variables, and introduces an independence loss designed to enforce the statistical factorization required for tractable Bayesian posterior estimation without repeated forward PDE solves (Guan et al., 2023). This is not a conventional PINN architecture, but it occupies the same inverse-problem landscape and addresses a key limitation of standard IPINNs: efficient uncertainty-aware posterior sampling.

Simulation-augmented training is represented by SimPINNs. There, the inverse problem is regularized by combining real observations with simulated data from an approximate forward model, yielding a hybrid loss

$k$3

The stated motivation is strong nonlinearity, non-injectivity, and scarcity of labeled observations; the reported orbit-restitution experiments show improved accuracy and robustness relative to standard PINNs (Besnard et al., 2023).

Uncertainty can also be represented non-probabilistically. Interval PINNs and fuzzy PINNs train one network for solution bounds and another for the parameter fields that realize those bounds, thereby avoiding Monte Carlo simulation and correlation-length specification while directly learning extremal input fields (Fuhg et al., 2021). This suggests that the inverse PINN paradigm is compatible with multiple inverse-problem epistemologies: deterministic, Bayesian, simulation-driven, interval, and fuzzy.

6. Limitations, training pathologies, and open directions

The literature presents IPINNs as effective, but not uniformly stable. A central recurring issue is multi-objective imbalance. When data, PDE residual, boundary, and initial losses differ strongly in scale or stiffness, optimization can neglect some constraints or overfit others. The inverse-Dirichlet weighting study explicitly analyzes this failure mode and attributes it to vanishing task-specific gradients under multi-scale dynamics, heteroscedasticity, stiffness, or catastrophic interference in sequential training (Maddu et al., 2021). The porous-transport and PBPK studies make a related practical claim: inverse convergence depends critically on adaptive or carefully tuned weighting of loss components and parameter gradients (Berardi et al., 2024, Wickramasinghe et al., 16 Sep 2025).

Identifiability and architecture dependence are also common concerns. SimPINNs explicitly note the ambiguity induced by non-injective forward operators and address it by adding simulated supervised pairs (Besnard et al., 2023). The peridynamics paper argues that standard activations can fail to recover physically meaningful kernels, whereas RBF activations introduce the symmetry and locality bias needed for the task (Difonzo et al., 2023). In dynamic elasticity, correct parameterization of $k$4 and hard boundary-condition enforcement are described as critical, with many alternative parameterizations failing to converge (Kag et al., 2023).

Measurement quality, data modality, and physical assumptions remain limiting factors. Inverse supersonic-flow reconstruction depends on partial boundary data, Schlieren-derived density gradients, entropy enforcement, and positivity constraints, and XPINNs are used because locally complex flow structures challenge single-network expressivity (Jagtap et al., 2022). Thermal diffusivity estimation from evaporative cryocooling is explicitly described as an ill-posed inverse problem because the excitation is long-pulse rather than impulsive; the IPINN formulation is introduced precisely because Parker-type formulas become unreliable under such conditions (Zhu et al., 13 Sep 2025). For unbounded domains, the boundary-free methodology is stated to work only when solutions stabilize at infinity, and its sampling strategy is tailored accordingly (Pérez-Bernal et al., 12 Dec 2025).

Taken together, these works suggest a consistent but technically nuanced picture. IPINNs are best viewed not as a single algorithm but as a family of inverse scientific machine-learning methods in which physics residuals, observational constraints, and trainable representations of unknown quantities are jointly optimized. Their effectiveness depends not only on the governing equations but also on loss design, architecture bias, parameterization, sampling, and the identifiability structure of the inverse problem itself.