Physics-Informed Pretraining Stage

• The method initializes models by minimizing PDE residuals and enforcing boundary/initial condition constraints, positioning parameters near physically admissible regions.
• It uses composite loss functions that integrate physics-based penalties to reduce iterations and improve solution accuracy across varied PDE problems.
• Empirical studies show significant error reduction and faster convergence, enabling effective transfer learning and robust performance in scientific machine learning.

A physics-informed pretraining stage is an initial optimization phase in the training of machine learning models for solving partial differential equations (PDEs), in which the model parameters are fit to satisfy physical laws—often in the absence of, or with minimal, labeled solution data. This approach encodes physical constraints into the learning process from the outset, typically through loss functions that directly penalize deviations from governing equations and boundary/initial conditions. The objective is to construct parameter or representation initializations that lie close to the feasible solution manifold, improve convergence rates, and/or bias the model toward physically consistent solutions. The physics-informed pretraining paradigm is central to scientific machine learning methods including PINNs (Physics-Informed Neural Networks), neural operators, and multimodal foundation models, as well as various initialization and transfer-learning strategies. Approaches vary in their architectural, algorithmic, and mathematical treatment, but are unified by the direct enforcement of analytic, symbolic, or discovered physical laws prior to, or as part of, task-specific fine-tuning.

1. Conceptual Foundations and Rationale

The physics-informed pretraining stage was motivated by the observation that random initialization of deep networks often yields function classes far from the physically admissible region, resulting in slow or unstable convergence, poor generalization, or even fundamental failure modes when addressing high-frequency, multi-scale, or ill-conditioned PDEs. In PINNs and foundation models, the incorporation of a physics-informed loss constrains the optimization trajectory to neighborhoods where physical laws are already satisfied—or nearly so—facilitating more effective learning when supervision is limited or the solution manifold is highly nonconvex (Mustajab et al., 2024, Cheng et al., 2024, Li et al., 27 Jan 2025, Zhu et al., 28 Dec 2025).

Empirical investigations confirm that physics-informed pretraining cuts iteration counts (often by 2× or more), sharpens final solution accuracy (by up to an order of magnitude), increases robustness to hyperparameters and data distribution, and enables rapid adaptation or fine-tuning to new solution regimes or PDE families with limited labeled data (Cheng et al., 2024, Zhu et al., 28 Dec 2025).

2. Mathematical Formulations of Physics-Informed Pretraining

Physics-informed pretraining generally involves minimizing a composite loss comprised of PDE residuals, boundary/initial condition mismatches, and, when relevant, supervised data terms. In neural PDE solvers, the pretraining loss for parameter vector $\theta$ often takes the following form: $$\mathcal{L}_\text{pretrain}(\theta) = \lambda_\text{res}\, \mathcal{L}_\text{residual} + \lambda_\text{IC}\, \mathcal{L}_\text{IC} + \lambda_\text{BC}\, \mathcal{L}_\text{BC} + \lambda_\text{data} \mathcal{L}_\text{data}$$ with terms

• $$\mathcal{L}_\text{residual}=\frac1{N_r}\sum_{i=1}^{N_r} \left[ R[u_\theta; x^{(r)}_i, t^{(r)}_i] \right]^2$$, $R$ being the analytic (or data-driven) PDE residual
• $\mathcal{L}_\text{IC}, \mathcal{L}_\text{BC}$: mean-squared errors on initial and boundary data, respectively
• $\mathcal{L}_\text{data}$: (optional) mean-squared error over labeled solution pairs

Variational approaches directly minimize the physics loss on collocation (unlabeled) points, e.g., as in PINNs, neural operators, and PFEM (Zhu et al., 28 Dec 2025, Wang et al., 6 Jan 2026, Mustajab et al., 2024).

Alternative methods employ hard constraints (e.g., trust-region SQP), mixed-integer linear programming for physics-constrained weight initialization (Li et al., 27 Jan 2025), or contrastive losses leveraging physical evolution dynamics in the latent space (Lorsung et al., 2024). Table 1 summarizes several representative loss structures.

Method	Physics-Informed Loss Structure	Notable Features
PINN	Residual + IC/BC mse (+ optional data)	Soft constraints, AD for derivatives
trSQP-PINN	$\|\text{constraints}(\theta)\|^2$ pretrain	Hard constraints, feasibility pretraining
PFEM	$\|K U^\theta - F\|^2$ (FEM residual)	Explicit FE diff., unstructured point clouds
PICL	Gen. contrastive loss over system trajectories	Latent-space physics alignment, coefficient-based
PI-MFM	Tokenized PDE residuals + BC + IC + data	Symbolic PDE input, vectorized derivative comp.

3. Algorithms and Architectures in Physics-Informed Pretraining

Example: Standard PINN Pretraining Loop

• Parameterize the surrogate as $u_\theta(x,t)$ (fully connected/tanh or other architectures)
• At each optimizer step:
  1. Sample interior, boundary, and initial collocation points
  2. Compute $$\mathcal{L}_\text{Stage1} = \mathcal{L}_\text{IC/BC}+\mathcal{L}_\text{PDE}$$
  3. Update $\theta$ via Adam or L-BFGS (Lin et al., 2021, Mustajab et al., 2024)

Advanced Pretraining Strategies

• Trust-region SQP-PINN: Pretrain $\theta$ to minimize constraint norm $\|c(\theta)\|^2$ with L-BFGS, outputting a nearly feasible initial point. Subsequent SQP iterations begin from this warm start, yielding halved outer iteration counts and order-of-magnitude error reduction in penalty/ALM variants (Cheng et al., 2024).

• MILP-based Initialization: Optimize the first layer of a PINN via boundary-only or physics-inclusive mixed-integer linear programming, guaranteeing deterministic, problem-aware initial weights. PINNs so-initialized converge in half the time and achieve 2–10× lower mean squared error post-training (Li et al., 27 Jan 2025).
• PFEM (Transolver Neural Operator): Match candidate solution $u_\theta(x)$ to PDE and BCs on unstructured meshes; finite element residuals computed explicitly. No solution labels are used; express feature encodings for geometry, materials, and BCs. Generalizes across mesh resolutions, with $O(1\%)$ relative error (Wang et al., 6 Jan 2026).
• Physics-Informed Contrastive Learning: During pretraining, anchor latent representations to match system evolution differences and impose PDE coefficient similarity in the contrastive loss, so the neural operator clusters representations by PDE family in its feature space (Lorsung et al., 2024).
• Physics-Informed Temporal Alignment: In foundation models with autoregressive prediction, enforce alignment between physical laws discovered on outcome trajectories and model roll-outs via data-driven inverse-problem losses, deterring shortcut behaviors (2505.10930).
• Multimodal/Symbolic Approaches (PI-MFM): Accept tokenized PDE descriptors as inputs, construct physics-informed objectives from parsed expressions, and pretrain with randomized collocation (Zhu et al., 28 Dec 2025).

4. Empirical Impact, Quantitative Comparisons, and Best Practices

Quantitative evidence strongly supports physics-informed pretraining:

• PINN pretraining with low-to-high frequency transfer learning reduces epoch counts to reach specified PDE-loss by $10$–$30\times$, achieves consistent $\sim1\%$ test errors at high frequencies, and enables convergence where vanilla PINNs fail (Mustajab et al., 2024).
• Trust-region SQP-PINN pretraining lowers absolute error from $8.953\times10^{-1}$ (no pretrain) to $0.364\times10^{-1}$ (with pretrain) on reaction-diffusion PDEs—a full order-of-magnitude improvement (Cheng et al., 2024).
• MILP-based initialization cuts full-solve wall times in half (from ~36min to ~20min) and reduces final MSE from $2.2\times10^{-5}$ (vanilla) to $0.8\times10^{-5}$ (boundary PT, 32 nodes). Full-PT is more costly without commensurate gains (Li et al., 27 Jan 2025).
• PFEM pretraining yields relative errors $e_\text{rel}\approx1.1\times10^{-2}$ and enables $5$–$9\times$ speedups in downstream FEM solvers compared to zero initialization (Wang et al., 6 Jan 2026).
• In multimodal foundation models, disabling physics losses increases mean $L^2$ error from $\sim3\%$ (with physics) to $\sim15\%$ (without), and physics pretraining improves out-of-domain robustness and data efficiency, especially in scarce-data regimes (Zhu et al., 28 Dec 2025).

Guidelines:

• Use Adam for pretraining, possibly switching to L-BFGS for fine loss minimization.
• Carry weights forward across frequencies or domains.
• Randomly resample collocation points per step to enhance generalization and lower error by up to $50\%$ (Zhu et al., 28 Dec 2025).
• For transfer learning or two-stage schemes, pretrain on initial/boundary tasks characteristic of the solution branch (e.g., match $u_0(x)$ before PDE enforcement) (Song et al., 2024, Lin et al., 2021).
• Use explicit hard constraints or feasibility loss when launching hard-constrained algorithms (Cheng et al., 2024).

5. Variants and Specializations

Two-Stage and Transfer-Learning Approaches

• Two-stage methods pretrain to fit e.g., an initial value or low-frequency solution, then fine-tune to enforce full physics—preventing trivial solutions and improving branch selection in nonlinear wave equations (Song et al., 2024, Lin et al., 2021, Mustajab et al., 2024).
• Transfer-learning schedules (e.g., training on $\omega_0=20,30,\ldots,60$ Hz sequentially) accelerate high-frequency convergence and prevent optimizer failures (Mustajab et al., 2024).

Contrastive, Multimodal, and Operator-Space Pretraining

• Contrastive learning in operator space, anchored with physics-based distance and coefficient similarity, enables multi-equation generalization and robust downstream task performance across Heat, Burgers, and Advection PDEs (Lorsung et al., 2024).
• Multimodal models (PI-MFM) accept formal PDE descriptors and initial data, assemble vectorized physics-informed losses, and demonstrate effective zero-shot adaptation (Zhu et al., 28 Dec 2025).

Model-Agnostic, Physics-Only, and Local Regression Alternatives

• Direct constraints-based regression (DCBR) eliminates all pretraining; single-point queries are solved via local constrained Taylor expansions and PDE equality constraints, with comparable accuracy to globally pretrained PINNs, but per-query (not amortized) solve time (Sabug et al., 15 Dec 2025).

6. Practical Implications and Limitations

The principal advantage of physics-informed pretraining is acceleration of convergence and assurance of physical plausibility without reliance on large labeled datasets, enabling practical learning in data-scarce, highly nonlinear, or multi-scale PDE settings. It is effective for both soft-constraint (PINN), hard-constraint (SQP), foundation model, and operator-learning scenarios (Zhu et al., 28 Dec 2025, Wang et al., 6 Jan 2026, Cheng et al., 2024).

Potential limitations include increased computational and memory cost (notably for large batch AD), sensitivity to pretraining sample or collocation point choice, and (for rigidity-inducing formulations) possible hindrance of adaptation to strongly data-driven features. Some variants, such as MILP-based initialization or explicit FEM-based loss, have scalability concerns for large models or very high-dimensional systems (Li et al., 27 Jan 2025, Wang et al., 6 Jan 2026). Inverse-problem-based self-supervision (e.g., PITA) requires careful handling of noise and underdetermined dynamics (2505.10930).

Pointwise regression without global pretraining (e.g., DCBR) eliminates training latency but incurs a higher per-query prediction cost, typically suitable for sparse, real-time, or interpretable settings (Sabug et al., 15 Dec 2025).

7. Connections to Broader Scientific Machine Learning Practice

Physics-informed pretraining unifies a large class of SciML approaches, aligning neural models, optimization-based solvers, operator networks, and hybrid workflows around the consistent enforcement of analytic or data-driven laws governing target processes. It underpins progress in neural surrogates for forward and inverse PDE problems, accelerates fine-tuning and transfer to unseen regimes, and enables robust prediction under data scarcity, distribution shift, or evolving physics. As models scale to foundation paradigms (PI-MFM), inclusion of physics losses in pretraining is foundational for durable, data-efficient generalization and rapid zero-shot adaptation (Zhu et al., 28 Dec 2025).

Fundamental advances continue in devising more scalable, robust, and interpretable pretraining losses; integrating contrastive, symbolic, and inverse-problem signals; efficiently encoding geometry/material/BC information; and coupling with classical solvers (e.g., PFEM) to realize the full potential of physics-informed learning.