Physics-Informed Loss Formulation

• Physics-informed loss formulation is a methodology that embeds differential equations and physical constraints into neural network training to ensure solutions respect underlying physics.
• It combines strong, weak, and hybrid approaches to balance pointwise fidelity with global regularization, achieving improved boundary enforcement and overall model robustness.
• Adaptive techniques like gradient-based weighting, dual-norm methods, and kernel-based losses optimize training stability and provide rigorous error estimates in scientific machine learning.

Physics-informed loss formulation is a core methodological scaffold in scientific machine learning, integrating domain-governing differential equations and physical constraints directly into the optimization objectives for neural architectures and operators. These loss functions drive the surrogate model to approximate solutions satisfying the prescribed physics, boundary, and initial conditions, while facilitating robustness, regularization, and, in some cases, certified error control. Modern formulations range from strong (pointwise) constraints, through weak and variational forms, to sophisticated hybrid, mixed, and kernel-based risk minimizations. The selection, implementation, and balancing of physics-informed losses are central to the accuracy and generalizability of machine learning solvers for PDEs and physics-constrained inverse problems.

1. Strong-Form (Collocation) and Weak-Form Losses

Strong-form (collocation) loss penalizes the squared residual of the PDE and boundary/initial conditions at a finite set of collocation points. For a generic PDE operator $\mathcal{L}[u](x)=0$ in a domain $\Omega$ with boundary operator $\mathcal{B}[u](x)=0$ on $\partial\Omega$, and with a trial network solution $u_\theta:\Omega\to\mathbb{R}$, the mean-squared error loss reads:

$$L_{\text{local}}(\theta) = \sum_{i=1}^{N_\Omega} |\mathcal{L}[u_\theta](x_i)|^2 + \sum_{j=1}^{N_{\partial\Omega}} |\mathcal{B}[u_\theta](x_j^\partial)|^2$$

Weak-form (variational) loss imposes the PDE in the sense of test functions via domain integrals or Petrov-Galerkin duality, often following an integration-by-parts procedure. For test functions $v_k \in V$:

$$L_{\mathrm{weak}}(\theta) = \sum_{k=1}^M \left| a(u_\theta, v_k) - l(v_k) \right|^2$$

where $a(\cdot, \cdot)$ is the bilinear form derived by IBP and $l(\cdot)$ is the corresponding linear functional, incorporating source and natural boundary data. The weak form is less sensitive to pointwise noise and provides a global constraint, mitigating overfitting at collocation points, enhancing generalization, and acting as a Sobolev-type global regularizer (Paine et al., 9 Feb 2026).

A hybrid loss is often constructed to combine both forms:

$\Omega$0

with tunable weights. This hybridization leverages the strengths of both approaches, yielding improved solution regularity, superior boundary propagation, and robustness against trivial or degenerate minima.

2. Adaptive and Robust Loss Balancing

Physics-informed training invariably introduces multiple competitive objectives: domain residual, boundary residual, initial condition enforcement, and (where applicable) data fidelity. Naive summation or fixed weighting of these loss terms can result in ill-posed or poorly-conditioned optimization, manifesting as non-convex loss landscapes, vanishing gradients, or an inability to simultaneously enforce all constraints (Basir et al., 2022, Bischof et al., 2021).

Modern balancing strategies include:

• Gradient-based weighting: Matching the gradient magnitudes (e.g., Learning Rate Annealing) or enforcing equal training rates (e.g., GradNorm).
• Self-adaptive scaling: SoftAdapt uses an exponential weighting based on recent loss changes.
• Relative loss balancing with lookback (ReLoBRaLo): Applies exponentially-smoothed, stochastically-referenced weight updates; especially effective in multi-term objectives and outperforms heuristic/manual balancing in a range of PDE benchmarks (Bischof et al., 2021).

Furthermore, augmented Lagrangian/PECANN approaches replace soft-penalty terms with equality constraints driven to zero via alternating Lagrange multiplier and primal variable updates, producing well-shaped single-basin landscapes and exact boundary satisfaction (Basir et al., 2022).

3. Alternative and Advanced Loss Formulations

a) Minimum-residual (dual-norm) formulations

Robust VPINN (RVPINN) adopts a minimum-residual principle in the discrete dual norm of the weak residual, guaranteeing an efficient estimator of the energy error and yielding loss metrics provably equivalent to the PDE solution error in the energy norm. These losses are basis-independent, robust to scaling, and provide rigorous a posteriori bounds (Rojas et al., 2023).

b) A posteriori error-majorant loss

The "Astral loss" optimizes a functional majorant (Repin-type error estimate) directly, allowing the network to simultaneously learn the primary solution and a flux field that together yield a guaranteed upper error bound. This approach is robust to solution singularities and provides practical certificates for neural operators and PINNs (Fanaskov et al., 2024).

c) First-order and mixed formulation losses

Rewriting higher-order PDEs as first-order systems—by introducing auxiliary variables for fluxes or derivatives—reduces backpropagation complexity, enables exact boundary enforcement via coordinate-adapted distance functions, and improves parameter identification accuracy. Mixed-formulation PINNs further support heterogeneous and coupled multiphysics systems, combining strong and weak terms and explicit constitutive relationships for primary and auxiliary unknowns (Gladstone et al., 2022, Harandi et al., 2023).

d) Single-term, causality-respecting losses

Recent works eschew separately-weighted initial/boundary/data losses by re-casting the entire PINN objective as a single residual term, using generalized functions (Heaviside, Dirac-delta) and appropriately regularized single-loss objectives that automatically encode IC/BC information and temporal causality (Es'kin et al., 2023).

4. Discretization-aware, External Solver, and Kernel-based Losses

Solver-coupled PINN losses integrate the discrete residuals from external finite-volume, finite-element, or reduced-order solvers as loss terms, enabling direct reuse of mature CFD codes and objective regularization in mesh-centric coordinates. A Jacobian-correction mechanism addresses non-differentiability issues, ensuring proper gradient propagation even when the residuals are not part of the autodiff graph (Halder et al., 29 Sep 2025, Mao et al., 2024).

Physics-informed kernel learning (PIKL): This framework replaces neural-function approximators with regularized kernel regressors, with the empirical risk

$\Omega$1

trading off data fidelity, Sobolev regularization, and PDE constraint. PIKL admits closed-form solutions, fast Fourier-based construction, and nonparametric error-rate guarantees (Doumèche et al., 2024).

5. Domain Decomposition and Interface Losses

To accommodate complex domains or parallelization, domain decomposition splits $\Omega$2 into $\Omega$3 subdomains, solving on each with local models $\Omega$4 and enforcing interface continuity via interface MSE penalties:

$\Omega$5

Total loss is the sum over all local and global (weak-form) losses plus interface continuity, weighted appropriately. This structure supports hybridization with quantum or classical differentiation and has seen application in quantum circuit-based variational machine learning (Paine et al., 9 Feb 2026).

6. Robustness, Generalization, and Loss Landscape

Physics-informed loss landscapes are, in many practical settings, locally smooth, strongly convex in the vicinity of the solution, and admit global mode-connectivity—a sharp contrast to the high non-convexity observed in other deep learning domains. Succeeding PINN and Deep Ritz (variational) losses share monotonic linear-interpolation properties and low-dimensional basins, although landscape conditioning may degrade with poor balancing or overly stiff PDEs (Rowan et al., 5 Feb 2026).

In weak-only or collocation-only regimes, pathologies include overfitting of pointwise data, numerical boundary issues, or susceptibility to trivial solutions. Hybrid, dual-norm, and global-regularized losses offer demonstrable increases in solution accuracy, improved boundary enforcement, and mitigation of spurious minimizers, as evidenced by multi-domain PDE and quantum problems (Paine et al., 9 Feb 2026).

7. Application-specific Physics-Informed Losses

Beyond generic PDEs, loss formulations have been specialized for:

• Tomographic inversion and image reconstruction: Physics-informed terms encoding line-integral (Radon) or projection operator constraints bring predicted images into consistency with forward measurement processes (Wang et al., 2024).
• Boundary-aware segmentation: Losses incorporating elastic-interaction energies derived from dislocation or material theories provide fine, physically-aligned boundary coherence in medical and remote-sensing segmentation tasks (Irfan et al., 25 Nov 2025).
• Material constraint regularization: Explicit addition of divergence or equilibrium residuals stabilizes inverse or supervised surrogates for mechanical problems, accelerates convergence, and reduces constraint errors (Lenau et al., 2024).

Table: Canonical Loss Types and Their Key Properties

Loss Type	Mathematical Character	Target Application / Advantage
Collocation (strong)	Pointwise MSE on DE/BC	Local fidelity, meshless solvers
Weak/variational	Integral, test-fn MSE	Global regularization, BC propagation
Hybrid	Weighted strong+weak	Combined accuracy and robustness
A posteriori bound	Majorant-based	Certified error, functional adaptivity
Dual-norm (RVPINN)	Min residual (dual)	Basis-independent, robust estimation
First-order/mixed	Reduced PDE system	Lower cost, exact BC, multiphysics
Kernel PIKL	RKHS-soaked quadratic	Closed-form solvers, theory-backed
Discretization loss	Mesh residual/solver	Direct reuse of FVM/ROM codes

Physics-informed loss formulation thus constitutes a rapidly-evolving and foundational component of machine learning for scientific, engineering, and inverse problems, with theoretical insight, practical implementation, and domain adaptation tightly intertwined (Paine et al., 9 Feb 2026, Rojas et al., 2023, Doumèche et al., 2024, Gladstone et al., 2022, Harandi et al., 2023, Akrivis et al., 2024).