Physics-informed Adaptation Learning

• Physics-informed Adaptation Learning is a family of methods that integrate known physical structures, such as measurement models and PDE operators, to guide the adaptation process.
• It employs diverse mechanisms including source-free test-time adaptation, meta-learning, and constraint-preserving techniques to optimize sensor and imaging systems.
• Empirical studies in EEG, microscopy, and cardiac MRI demonstrate significant performance gains by aligning adaptation with domain-specific physics.

Physics-informed Adaptation Learning denotes a family of methods in which adaptation is organized around known physical structure rather than purely statistical alignment. In the current literature, that structure may be a measurement model, a forward operator, a signal-generation mechanism, a diagnostic geometry, a conservation law, a variational condition, a latent state-space model, or the kernel of a differential operator. Taken together, the literature uses the phrase across several non-equivalent regimes: unsupervised domain adaptation across heterogeneous sensors, source-free test-time adaptation, meta-learning over families of PDEs and dynamical systems, adaptive self-supervision in PINNs, and post-hoc constraint adaptation that preserves previously learned physics (Mellot et al., 2024, Nguyen et al., 7 Jul 2025, Li et al., 29 Oct 2025, Torres et al., 23 Mar 2025).

1. Conceptual scope

A defining feature of the area is that adaptation is moved upstream toward the physics of acquisition or dynamics. In heterogeneous EEG, adaptation is performed first at the signal level by interpolating all recordings onto a common virtual montage using source-model-based field interpolation, and only then at the covariance-manifold level by Riemannian re-centering (Mellot et al., 2024). In optical microscopy of 2D materials, adaptation is framed as inverting nuisance image-formation factors such as white balance and spectral response, recovering reflectance, and re-rendering target images into the synthetic source style before prediction (Nguyen et al., 7 Jul 2025). In multiparametric cardiac MRI, test-time adaptation is guided by synthetic reference series generated from fitted signal equations rather than by generic image similarity alone (Li et al., 29 Oct 2025).

The term also extends beyond domain adaptation in the narrow machine-learning sense. Adaptive PINN work treats collocation placement, task weighting, grid refinement, and learned optimization as adaptation variables because they determine where and how physics is enforced during training (Subramanian et al., 2022, Nguyen et al., 2022, Li et al., 28 Sep 2025, Rigas et al., 2024, Boudec et al., 2024). Other work recasts adaptation as fast retuning across task families: a learned solver updates PDE coefficients by transforming the physics-informed gradient in a task-conditioned way (Boudec et al., 2024), while Pi-PINN and related acoustic models replace iterative fine-tuning by closed-form output-head adaptation under operator constraints (Wong et al., 23 Apr 2026, Komaba et al., 7 Jun 2026).

A recurring ambiguity is that not every paper uses “adaptation” in the same way. The Onion model for fusion diagnostics is explicitly presented as a transferable physics-informed module pattern across backbones and devices, but not as formal unsupervised domain adaptation (Wang et al., 2024). The adaptive PINN survey correspondingly treats transfer learning and meta-learning over PDE instances as the central adaptation paradigm for PINNs, rather than source-target distribution alignment in the classical vision sense (Torres et al., 23 Mar 2025).

2. Major methodological families

The literature clusters into a small number of recurring methodological families.

Family	Representative papers	Core adaptation variable
Measurement-space harmonization	EEG FI, $\varphi$-Adapt, PI-TTA MRI (Mellot et al., 2024, Nguyen et al., 7 Jul 2025, Li et al., 29 Oct 2025)	common sensor/image representation
Task-family transfer and fast retuning	reactor meta-learning, neural solver, Pi-PINN, acoustic PIELM (Wang et al., 2024, Boudec et al., 2024, Wong et al., 23 Apr 2026, Komaba et al., 7 Jun 2026)	initialization, optimizer, or linear head
Adaptive supervision in PINNs	Adaptive-G/Adaptive-R, FBOAL, AW-EL-PINNs, PIKAN (Subramanian et al., 2022, Nguyen et al., 2022, Li et al., 28 Sep 2025, Rigas et al., 2024)	collocation set, loss weights, grid state
Constraint-preserving post-hoc adaptation	operator-invariant $d_{\text{eff}}$ and null-space projection (Otchere et al., 4 Jun 2026)	PDE-null parameter subspace
Physics-weighted source selection or context adaptation	PI-MSDA, SeqBattNet (Zhang et al., 30 Sep 2025, Tran et al., 22 Sep 2025)	source weights or cycle-specific physical parameters

What unifies these families is not a single optimization recipe but a common design principle: adaptation is constrained to pass through variables that are physically interpretable or operator-compatible. In some cases the physical carrier is explicit measurement geometry, as with EEG lead fields and fusion line-integral contribution matrices (Mellot et al., 2024, Wang et al., 2024). In others it is a hidden but structured state, such as battery SOH, RC time constants, or reactor kinetics encoded through meta-learned recurrent dynamics (Tran et al., 22 Sep 2025, Wang et al., 2024). In adaptive PINNs, the physical carrier may instead be the spatial support of the PDE loss or the dimensionality of the operator kernel (Subramanian et al., 2022, Otchere et al., 4 Jun 2026).

3. Physics carriers and adaptation variables

The most distinctive aspect of the field is the variety of objects that carry the physics prior.

In EEG, the carrier is the quasi-static electromagnetic forward/inverse relation between latent neural generators and scalp potentials. The proposed field interpolation constructs a linear operator

$\hat{X}=AX,$

mapping a dataset-specific montage to a fixed 17-channel template before covariance computation, thereby resolving feature-space mismatch induced by heterogeneous caps (Mellot et al., 2024). The subsequent covariance representation lies on the SPD manifold and is adapted by Riemannian re-centering,

$$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$

In $\varphi$-Adapt, the carrier is the optical image-formation model. Synthetic images are generated as

$x_s=A_sR_s,$

while real target images are modeled as

$x_t=\operatorname{diag}(G_t)A_tR_t.$

Adaptation therefore attempts to estimate and undo the unknown $G_t$ and $A_t$, recover $R_t$, and synthesize a source-style image

$d_{\text{eff}}$0

The learnable modules ColorNorm and SpecInv are organized around these physical variables rather than around an unconstrained latent style space (Nguyen et al., 7 Jul 2025).

In cardiac MRI, the carrier is the signal equation of the target acquisition. For T1 inversion recovery, the synthetic reference used during test-time adaptation is generated from fitted parameters through

$d_{\text{eff}}$1

The adaptation target is not a generic registration template but a subject-specific motion-free series synthesized from the fitted physics model (Li et al., 29 Oct 2025).

In fusion diagnostics, the carrier is device geometry. Onion ingests contribution matrices $d_{\text{eff}}$2 encoding line-of-sight geometry and fuses geometry features with data features multiplicatively, rather than by addition or concatenation, so that static physical information modulates the dynamic measurement stream (Wang et al., 2024). In structural-health monitoring, the carrier is simpler: PI-MSDA weights source domains by a Gaussian-kernel similarity computed from a “key physics” descriptor, instantiated as relative floor height in the case study (Zhang et al., 30 Sep 2025).

In PDE-oriented adaptation, the carrier can be the operator itself. Pi-PINN learns a shared embedding $d_{\text{eff}}$3 across related PDE instances and adapts a new task by solving a linear head

$d_{\text{eff}}$4

where $d_{\text{eff}}$5 is assembled from PDE, BC, and IC constraints applied to the embedding (Wong et al., 23 Apr 2026). The effective-dimensionality framework goes further by defining

$d_{\text{eff}}$6

and interpreting the remaining unconstrained directions as the adaptation capacity left after PDE pretraining (Otchere et al., 4 Jun 2026).

4. Adaptation mechanisms

Several adaptation mechanisms recur across the literature.

A first class performs explicit source-free or unsupervised adaptation on unlabeled target data. In $d_{\text{eff}}$7-Adapt, the adaptation signal is prediction entropy,

$d_{\text{eff}}$8

and the updated parameters are specifically the physics-informed front-end modules $d_{\text{eff}}$9 and $\hat{X}=AX,$0, not the classifier head (Nguyen et al., 7 Jul 2025). In EEG, unlabeled target data are used to estimate the target-domain Riemannian mean for whitening, with first session, first run, or first-half data depending on the dataset structure (Mellot et al., 2024).

A second class uses transductive test-time adaptation. Cardiac MRI PI-TTA first applies a contrast-agnostic pretrained groupwise registration network, then fits a signal model to the warped sequence, generates a synthetic reference, and fine-tunes the network for 10 steps on that individual test case (Li et al., 29 Oct 2025). SeqBattNet similarly infers cycle-specific adaptation parameters once from the initial prefix of a discharge cycle and then rolls out a mechanistic decoder for the remainder of that cycle (Tran et al., 22 Sep 2025).

A third class performs task-family adaptation through meta-learning or learned optimization. The reactor foundation-model framework meta-trains a recurrent network over 1500 simulated tasks from CSTR, batch, and plug flow reactors using Reptile, then adapts to an unseen reactor with few-shot data and a physics-informed loss (Wang et al., 2024). The neural parametric solver replaces fixed optimization rules by

$\hat{X}=AX,$1

so the update itself is conditioned on the current PDE instance (Boudec et al., 2024).

A fourth class adapts the supervision pattern rather than the model output directly. Adaptive-G and Adaptive-R periodically reallocate a fixed collocation budget toward regions identified by the PDE residual or the gradient of the PDE loss, with a cosine-annealed mixture of uniform and adaptive samples (Subramanian et al., 2022). FBOAL similarly performs fixed-budget replacement of low-residual collocation points by high-residual candidates identified locally on subdomains, thereby redistributing effort across space, time, and parameter values without growing the dataset (Nguyen et al., 2022). AW-EL-PINNs adapt the relative importance of Euler–Lagrange residual terms through trainable uncertainty-style weights $\hat{X}=AX,$2, while adaptive PIKAN training coordinates grid extension, optimizer-state interpolation, residual-based attention, and residual-based sampling (Li et al., 28 Sep 2025, Rigas et al., 2024).

A fifth class is explicitly constraint-preserving. After PDE pretraining, the effective-dimensionality paper computes the projector

$\hat{X}=AX,$3

onto the PDE-null parameter subspace, forms boundary-adaptation directions inside that null space, and updates only along those directions so that new BCs or ICs can be satisfied without disturbing the learned physics to first order (Otchere et al., 4 Jun 2026).

5. Applications and empirical findings

The empirical record is heterogeneous but already broad.

Application	Reported finding	Paper
Heterogeneous EEG adaptation	FI consistently outperforms other methods when few channels are shared; significantly higher accuracy than SSI on four datasets with $\hat{X}=AX,$4	(Mellot et al., 2024)
2D quantum flakes	thickness error $\hat{X}=AX,$5 nm; $\hat{X}=AX,$6 nm lower than the previous domain adaptation method compared against	(Nguyen et al., 7 Jul 2025)
Fusion line-integral diagnostics	physics input reduces $\hat{X}=AX,$7 by approximately $\hat{X}=AX,$8 on synthetic and about $\hat{X}=AX,$9 on experimental data	(Wang et al., 2024)
Adaptive PINN collocation	vanilla PINNs can reach up to $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$0 prediction error; adaptive schemes achieve up to an order of magnitude smaller error	(Subramanian et al., 2022)
Transferable PINN heads	prediction/adaptation $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$1–$$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$2 faster than a typical PINN	(Wong et al., 23 Apr 2026)
Acoustic field adaptation	$$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$3 s adaptation versus $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$4 s for PINN fine-tuning with comparable interpolation accuracy	(Komaba et al., 7 Jun 2026)

In EEG, the central empirical regime is sparse channel overlap. Across six public motor-imagery datasets, only Cz is common to all montages; field interpolation therefore addresses a regime in which common-channel selection is structurally weak. The reported advantage is strongest when the shared channels are few or anatomically misaligned, exactly the case targeted by the physics-informed interpolation stage (Mellot et al., 2024).

In 2D-material microscopy, the empirical gains are strongest for thickness estimation. The synthetic generator produces 600,000 images with approximately one million instances, and $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$5-Adapt reports a thickness error of 5.8 nm, compared with 25.3 nm for source only and 14.9 nm for the strongest listed source-free baseline. The ablation study further shows that entropy minimization alone helps, but ColorNorm, Source Transform, and the wavelength-neighbor regularizer each add further gains (Nguyen et al., 7 Jul 2025).

In fusion diagnostics, the picture is more nuanced. Geometry-aware feature injection and Softplus improve both $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$6 and $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$7, especially on synthetic phantom data, but the physics-informed reprojection loss can deliberately worsen label agreement while improving forward-consistency with the measured line integrals. This reflects the fact that real labels are themselves outputs of inversion codes and may not be physically consistent with the raw measurements (Wang et al., 2024).

For PDE-family adaptation, fast head adaptation and learned solvers shift the efficiency frontier. Pi-PINN reports fast and accurate physics-informed solutions for unseen PDE instances without requiring data for those instances (Wong et al., 23 Apr 2026). The acoustic PINN–PIELM hybrid shows that physics-constrained output-layer adaptation can nearly match moderate PINN fine-tuning while reducing adaptation time by more than three orders of magnitude (Komaba et al., 7 Jun 2026). In constraint adaptation, subspace projection reaches near-equivalent boundary-fit quality in seconds to minutes, whereas full gradient-based fine-tuning can match or exceed it only with substantially მეტი wall-clock time and tuning (Otchere et al., 4 Jun 2026).

6. Limitations, ambiguities, and research directions

The field’s main limitation is conceptual heterogeneity. “Physics-informed adaptation learning” does not yet denote a single canonical problem class. It can refer to source-free visual adaptation, transductive test-time fine-tuning, meta-learned PDE adaptation, collocation adaptation inside PINN training, or null-space constraint retuning after pretraining (Nguyen et al., 7 Jul 2025, Li et al., 29 Oct 2025, Torres et al., 23 Mar 2025, Otchere et al., 4 Jun 2026). This suggests that the term currently functions more as a design philosophy than as a settled taxonomy.

A second limitation is dependence on the adequacy of the physics prior. EEG field interpolation assumes that scalp voltages are generated by common latent cortical sources through quasi-static electromagnetic propagation (Mellot et al., 2024). $$C_i^{(\mathrm{rct})}=\bar{C}^{-\frac12}C_i\bar{C}^{-\frac12}.$$8-Adapt models optical properties but explicitly does not account for sensor noise and artifacts (Nguyen et al., 7 Jul 2025). MRI PI-TTA requires a reasonably accurate signal equation for the target acquisition and stable curve fitting after initial warping (Li et al., 29 Oct 2025). In PI-MSDA, the explicit prior used in experiments is only relative floor height, and the paper itself notes that richer descriptors such as modal frequencies, stiffness and mass distributions, damping, or sensor layout would be more informative (Zhang et al., 30 Sep 2025).

A third limitation concerns what is actually adapted. Some methods remain supervised within each device or dataset even when they are reusable across setups, as in Onion (Wang et al., 2024). Others are only locally adaptive: the effective-dimensionality method relies on local linearization around a pretrained solution, and its null-space interpretation is exact only under sufficient resolution and favorable numerical rank decisions (Otchere et al., 4 Jun 2026). Closed-form head adaptation methods are fast because only a linear head is adapted, but that speed is purchased by assuming that the shared embedding spans the relevant solution family (Wong et al., 23 Apr 2026, Komaba et al., 7 Jun 2026).

A recurring misconception is that physics-informed adaptation must appear as a residual penalty in the loss. The literature shows a much broader design space: physics can appear as a source model, a synthetic reference generator, a geometry tensor, a mechanistic decoder, a task-similarity weight, a collocation redistribution rule, or an operator-null projector (Mellot et al., 2024, Li et al., 29 Oct 2025, Wang et al., 2024, Tran et al., 22 Sep 2025, Nguyen et al., 2022, Otchere et al., 4 Jun 2026). Another misconception is that adaptation is always domain adaptation. The PINN survey instead centers transfer learning and meta-learning across PDE instances, and several adaptive PINN papers adapt supervision or optimization rather than source-target distributions (Torres et al., 23 Mar 2025).

The strongest research directions already articulated in the literature are consistent across domains: richer and more realistic forward models, uncertainty estimation, better benchmark standardization, adaptation under stronger distribution shift, broader cross-device or cross-lab studies, and methods that retain the computational advantages of fast adaptation without sacrificing the physical guarantees of mechanistic models (Nguyen et al., 7 Jul 2025, Wang et al., 2024, Banerjee et al., 2023, Torres et al., 23 Mar 2025). Taken together, these works suggest that the central challenge is no longer whether physics can be added to adaptation, but how to decide which part of the adaptation pipeline should be physically constrained: the representation, the optimization, the sampling distribution, the latent state, or the permissible update subspace.