Explicit Physics Optimizer

• Explicit physics optimizers are methods that leverage physical structures, such as stiff Jacobians and loss scaling, directly in update rules to improve numerical stability.
• They encompass various designs including half-inverse gradient, PDE-aware, and meta-learned neural optimizers, each tailored to handle stiffness, ill-conditioning, and loss imbalance.
• By incorporating governing physical equations, these optimizers reduce common challenges in machine learning applications, achieving superior convergence and reliable error minimization.

An explicit physics optimizer is any optimization method that incorporates physical structure, numerical characteristics, or analytic principles of a problem’s governing equations directly into its update rules or iterative scheme. This class encompasses gradient-based, quasi-Newton, Hamiltonian, meta-learned, and hardware-inspired optimizers, provided the optimization process leverages explicit knowledge of physics—such as stiff Jacobians, loss scaling from differential equations, constraint structure, or dynamical analogs. The foundational insight is that incorporating these structures—rather than relying on generic optimism from neural net training—alleviates issues such as ill-conditioned gradients, stiff modes, and imbalance between loss terms, which commonly arise in physics-informed machine learning, scientific computing, and experimental design.

1. Foundational Principles of Explicit Physics Optimization

Explicit physics optimizers are grounded in the exploitation of physics-derived information to tailor the search trajectory toward more robust and efficient convergence. Classical gradient-based optimizers—such as SGD or Adam—are often ill-suited for physical systems whose parameter-to-observable map may be highly stiff, ill-conditioned, or nonlinear due to underlying physical laws.

A prototypical example is the half-inverse gradient (HIG) method (Schnell et al., 2022), which addresses the mismatch between network parameter updates and the physical scaling of gradient directions, especially in coupled neural–physics systems. Here, the optimizer update is structured as

$\Delta\theta_{HIG} = -\eta J^{-1/2} \nabla_y L,$

where $J = \partial\phi/\partial\theta$ is the Jacobian of the physical solver with respect to parameters $\theta$. HIG interpolates between first-order gradient descent (no inversion, $\kappa = 1$) and Gauss--Newton (full inversion, $\kappa = -1$), achieving superior stability on stiff problems through partial Jacobian inversion.

In physics-informed PINNs, explicit optimizers correct for gradient conflicts and imbalance by parameterizing update rules on loss splitting, per-residual scaling, or constrained Lagrangian flows, further detailed in the subsequent sections.

2. Architecture of Update Rules: Classes of Explicit Physics Optimizers

Explicit physics optimizers encompass several algorithmic archetypes:

• Half-Inverse Gradient and Related Jacobian-Aware Methods: These precondition updates with a fractional inverse of the solver–network Jacobian, as in HIG, normalizing parameter directions with respect to the underlying physics. This corrects for a broad singular value spectrum in $J$ while avoiding the artifact amplification of full Newton steps (Schnell et al., 2022).
• Physics-Informed/Physical-Structure Adapted Adam Variants:
  • PDE-Aware Optimizer (Shukla et al., 10 Jul 2025): Adjusts per-parameter update size based on the variance of per-sample PDE residual gradients, enabling resolution of sharp spatial features by adaptively shrinking or growing update magnitudes.
  • MultiAdam (Yao et al., 2023): Maintains independent first and second moments for multiple grouped losses (e.g., boundary, PDE residuals), aggregating parameter updates in a scale-invariant manner to balance disparate physical loss scales.
  • Velocity-Regularized Adam (VRAdam) (Vaidhyanathan et al., 19 May 2025): Introduces a quartic kinetic energy term into Adam’s parameter space dynamics, providing velocity-dependent damping that stabilizes training near sharp minima, echoing dynamical behavior in physical systems.
• Hamiltonian, Lagrangian, and Dynamical-System Optimizers:
  • Energy-Conserving Descent (ECD) (Luca et al., 30 Jan 2025): Treats parameter evolution as a dissipative Hamiltonian system with a tunable microcanonical measure concentrated near the minimum of a loss, providing run-to-run stability and independence from initialization by ergodicity and energy conservation.
• Meta-Learned Neural Optimizers:
  • Metamizer (Wandel et al., 2024): Implements a U-Net neural network that takes as input the normalized current and previous gradients, as well as the last update, producing an update direction and step size that are scale-invariant and empirically robust across a range of PDEs.
• Augmented-Lagrangian and Hardware-Physical Solvers:
  • Lagrange Multiplier Physical Machines (Vadlamani et al., 2020): Utilize physical analogs of entropy-minimizing dissipative networks to realize constrained minimization, with pump gains serving as physical Lagrange multipliers.
  • Augmented Lagrangian Methods (for e.g., kinematic mass minimization) (Cho et al., 2015): Sequentially refine unconstrained subproblems by explicit penalization of constraint violation, tuned for complex physical event analyses.
• Physics-informed Surrogate and Bayesian Optimization:
  • Physics-informed Gaussian Process Optimizer (Hanuka et al., 2020): Embeds the Hessian of a fast surrogate physics model into the kernel of a GP, yielding strongly anisotropic exploration and rapid convergence during online optimization of experimental setups.

3. Theoretical Properties and Analytical Insights

A defining feature of explicit physics optimizers is their capacity to address ill-conditioning, loss scale disparity, and nonconvexity inherent to physical PDEs and inverse problems. For example, HIG can be interpreted as steepest descent under a metric that partially normalizes the singular directions of $J$, yielding step directions with norm $\|J^{3/4} v\|_2$ (Schnell et al., 2022). This enables simultaneous adaptation to large- and small-scale physical modes without instabilities typical of second-order Newton-type methods.

Convergence guarantees for some variants (e.g., MultiAdam) arise from theoretical analysis that maps $L^2$ loss reduction to pointwise error bounds in the PINN context, rendering the optimizer nearly invariant to domain scaling and robust under PDE loss/boundary loss imbalance (Yao et al., 2023). Physics-based Hamiltonian optimizers, such as ECD, explicitly generate a microcanonical measure peaked at the loss minimizer, with concentration controlled by an analytically tractable parameter $\eta$ (Luca et al., 30 Jan 2025). In all cases, the physical insight guides either the preconditioning (via Jacobian structure or loss-group scaling), the update metric, or the exploration-exploitation strategy (as in Bayesian GP kernels).

Limitations are domain- and architecture-dependent; HIG requires $O(p B^2 m^2)$ for SVDs of Jacobians (Schnell et al., 2022), PDE-Aware Optimizer may require careful tuning of variance term stabilizers for highly nonuniform solutions (Shukla et al., 10 Jul 2025), and hardware analog approaches hinge on realizable time-scale separations and physical constraint enforcement (Vadlamani et al., 2020).

4. Algorithmic Realizations: Pseudocode and Workflow Summaries

Optimization workflows for explicit physics optimizers are characterized by their explicit integration of physics operators or statistics:

• Half-Inverse Gradient (HIG, (Schnell et al., 2022))
  1. Evaluate all outputs and losses for the minibatch.
  2. Autodifferentiate to obtain loss gradients and Jacobian blocks $\partial y/\partial\theta$.
  3. Stack Jacobians and gradients, perform SVD.
  4. Form matrix power $J^{-1/2}$, truncating small singular values.
  5. Compute and apply the update $\Delta\theta = -\eta J^{-1/2} g$.

• PDE-Aware Optimizer (Shukla et al., 10 Jul 2025)

  1. Sample collocation points; compute per-sample PDE-residual gradients $g_i$.
  2. Update moving averages for first (mean) and second (elementwise squared) moments.
  3. Compute preconditioned step: divide mean by square-rooted variance vector.
  4. Update parameters using this preconditioned direction.

• MultiAdam (Yao et al., 2023)

  1. For each distinct loss group, compute gradients and moving moment estimates.
  2. Normalize each group’s update by its own second moment.
  3. Aggregate normalized group updates (arithmetic mean).
  4. Apply the resulting update to parameters.

• Energy-Conserving Descent (Luca et al., 30 Jan 2025)

  1. Evolve $\theta, p$ by Hamiltonian dynamics, with unit-norm velocity and an exponential transform of the network loss in the conserved quantity.
  2. Implement discrete update with (optionally) small stochastic "bounces" to ensure ergodicity and robust phase-space exploration.
  3. Final $\theta$ values approach the microcanonical distribution peaked at the loss minimum.

• Metamizer (Wandel et al., 2024)

  1. Collect normalized (scale-invariant) gradient, previous gradient, and previous update.
  2. Feed concatenated fields into a U-Net to obtain a direction and globally pooled scale.
  3. Apply per-cell update, proceed to next iteration.
  4. Meta-optimize U-Net weights over a pool of solution instances by minimizing propagated PDE residuals.

5. Empirical Results Across Benchmarks

Explicit physics optimizers consistently outperform generic methods in physics-driven tasks characterized by stiffness, multiscale effects, or loss imbalance:

• HIG achieves faster and more reliable convergence than Adam or Gauss–Newton in nonlinear oscillator control, Poisson, and quantum dipole systems, achieving loss plateaus orders of magnitude lower than Adam and greater uniformity across physics modes (Schnell et al., 2022).

• PDE-Aware Optimizer yields smoother convergence, lower relative $L^2$ errors (e.g., $1.1 \times 10^{-3}$ for Burgers, $1.5 \times 10^{-2}$ for Allen–Cahn), and suppresses error spikes in steep-gradient regions compared to Adam and Soap (Shukla et al., 10 Jul 2025).
• MultiAdam demonstrates 1–2 orders of magnitude improvement in accuracy over Adam and strong insensitivity to domain rescaling or loss weight tuning—e.g., reducing Helmholtz equation error from 22.5% (Adam) to 0.43% (Yao et al., 2023).
• Metamizer achieves near machine-precision residuals (e.g., $6.7 \times 10^{-33}$ on the Laplace equation), outpaces classical sparse solvers, and generalizes to unseen equation classes without retraining (Wandel et al., 2024).
• Lagrange Multiplier Physical Machines (OPO, RF oscillator arrays, polariton condensates) solve large combinatorial problems such as Ising energy minimization at rates and energy costs far below digital algorithms (Vadlamani et al., 2020).

6. Integration, Limitations, and Future Directions

Explicit physics optimizers are broadly compatible with autodiff frameworks and hardware acceleration. HIG and MultiAdam can be integrated into PyTorch, TensorFlow, or JAX via vector–Jacobian operations and differentiated physics residuals. Efficient SVD computation and streaming accumulation strategies are advised for scalability (Schnell et al., 2022, Shukla et al., 10 Jul 2025). Truncation thresholds, step-size decay rates, and update frequency of physical statistics are key hyperparameters tuned to the characteristic scales of the problem.

Major current limitations include computational bottlenecks for very high-dimensional output mappings (HIG, MultiAdam), overhead in meta-optimization training (Metamizer), and the requirement for actionable fast surrogate models in Bayesian-optimization contexts (Wandel et al., 2024, Hanuka et al., 2020). Open directions include scalable approximate SVDs, adaptive meta-optimizers for mixed mesh/topology scenarios, and real-time onboard hardware implementations for laboratory and industrial applications.

Explicit physics optimization thus serves as both a methodology and a design paradigm, incorporating mathematical structure from governing equations and physical laws directly into the iterative logic of machine learning, numerical analysis, and analog/hybrid computation. In doing so, these optimizers achieve performance and stability unattainable by architecture-neutral algorithms across a broad spectrum of scientific and engineering problems.