Physics-Informed Multi-Task Learning

• Physics-informed multi-task joint learning is a deep learning framework that integrates physical constraints with multiple related tasks to enhance predictive consistency and efficiency.
• It employs strategies like adaptive weighting, hard and soft parameter sharing, and cross-task information transfer to balance data and physics losses.
• The approach is validated in domains such as biomechanics, spatiotemporal dynamics, and optimal control, demonstrating significant improvements in accuracy and robustness.

Physics-Informed Multi-Task Joint Learning refers to deep learning methodologies that simultaneously address multiple related prediction or inference tasks, leveraging physical laws as inductive biases or constraints within a unified optimization framework. This paradigm has gained prominence where physical models co-exist with high-dimensional, multimodal, or sparse data, such as in biomechanics, spatiotemporal physics, optimal control, and systems biology. The joint training of several predictive or generative modules, each informed by physics, provides synergy: tasks benefit mutually from representation sharing, physics-constrained regularization, and adaptive cross-task information transfer, leading to improved accuracy, robustness, and scientific consistency.

1. Foundational Principles

Physics-informed multi-task joint learning extends classical PINNs by integrating several closely-related supervised, regression, or generative tasks, subject to physical constraints such as ODEs, PDEs, conservation laws, or system dynamics. The governing objective function for such frameworks generally takes the compositional form: $$\mathcal{L}_{\mathrm{total}}(\theta) = \sum_{k=1}^M \lambda_k\,\mathcal{L}_k(\theta)$$ where each $\mathcal{L}_k$ encodes a task-specific error or residual (data mismatch, physics residual, boundary constraint, etc.), and $\lambda_k$ is a (learned or tuned) balancing weight. Multi-task formulations support both hard parameter sharing—where a neural “trunk” learns common features with task-specific “heads" (e.g., state, control, costate in optimal control), and soft sharing via cross-attention, transfer learning, or adaptive task mixing.

Physics-informed constraints may be embedded in several ways:

• As residuals in the loss to enforce dynamic equilibrium, energy conservation, or admissibility of solutions (e.g., inverse dynamics for biomechanics, reaction–diffusion for cardiac dynamics).
• As part of the generative process or adversarial discrimination (e.g., physics-informed GANs for trajectory synthesis).
• As regularization within probabilistic/Bayesian inference, either as prior terms in the posterior or as soft penalty on model outputs.

2. Exemplary Frameworks and Methodologies

Physics-informed multi-task joint learning encompasses a broad spectrum of methodologies, with the following archetypes:

a) Multi-Resolution, Multi-Task RNNs:

In musculoskeletal inference, Taneja et al. implement a multi-resolution physics-informed RNN (MR PI-RNN) where a sequence of GRUs is trained recursively across coarse-to-fine signal scales, simultaneously predicting joint motion (kinematics) and identifying physiological parameters by enforcing dynamic equilibrium via a physics loss (Taneja et al., 2023).

b) Multi-Head PINNs for PDE Systems:

MH-PINNs utilize a shared feature extractor (“body”) parameterized by $\theta$ and individual linear “heads” $W_i$ for $T$ related physics-informed regression tasks. All tasks are regularized by their PDEs and data-fitting criteria; the joint loss aggregates over tasks, and uncertainty quantification is enhanced via normalizing-flows over task parameters (Zou et al., 2023).

c) Multi-Physics GANs for Trajectory Synthesis:

In arterial traffic, a coupled GAN framework jointly models lane-change events (LC-GAN) and car-following trajectory dynamics (Trajectory-GAN). Both GANs are trained under a joint objective, tightly integrating data-driven and physics-based features (IDM/MOBIL car-following priors, signal phase, geometric masks) (Xu et al., 20 Nov 2025).

d) Joint Biomechanical Modeling:

PiGRN and PI-MJCA-BiGRU architect multi-task deep recurrent models for biomechanics that simultaneously predict joint angles, velocities, accelerations, external loads, and muscle torques, embedding multi-joint rigid-body dynamics and physiological constraints into the loss (Kumar et al., 29 Aug 2024, Ma et al., 14 Nov 2025).

e) Multi-Task Gaussian Process Models:

In irregular spatiotemporal domains, P-M-GP constructs a multi-output GP prior over variables with Kronecker-structured kernel (task × spatial × temporal) and enforces governing PDE residuals as soft constraints in the joint marginal likelihood (Zhang et al., 15 Oct 2025).

f) Multi-Task PINNs for Optimal Control and Inverse Problems:

AW-EL-PINNs embed the Euler–Lagrange two-point boundary-value system as parallel tasks (state, costate, optimality), employing adaptive weighting derived from maximum likelihood to dynamically balance the tasks during training (Li et al., 28 Sep 2025). Pinning the tasks with data loss, boundary condition error, PDE residuals, and with auxiliary knowledge, is also demonstrated in structural inverse problems using transfer-learned multi-task PINNs (Xu et al., 2022).

3. Adaptive Weighting and Optimization Strategies

A critical challenge in multi-task, physics-informed learning is addressing loss imbalance—most physical and data losses differ by orders of magnitude or in their statistical properties, leading to optimization inefficiency or degraded solutions.

Adaptive Weighting:

• The AW-EL-PINNs introduce an adaptive weighting mechanism, with weights parameterized as $\lambda_i=e^{-s_i}$ and regularized, so that during training, losses “self-balance” according to uncertainty (Li et al., 28 Sep 2025).
• Bayesian approaches for PINNs propose setting the weights $w_k=1/\sigma_k^2$ where $\sigma_k$ reflect task noise, and further adapt these by matching gradient variances (“Inverse-Dirichlet” adaptation) across tasks (Perez et al., 2023). This mechanism is embedded within HMC-based sampling, stabilizing training and obtaining well-calibrated uncertainty estimates.

Multi-Objective Optimization:

• Pareto-optimal training via vectorized multi-objective loss and “gradient surgery” (projection of task gradients to avoid conflicts) ensures that no task decrease comes at the expense of another and avoids failure modes of scalar-weighted sums (Bahmani et al., 2021).

Knowledge Transfer and Representation Sharing:

• Multi-task frameworks leverage transfer learning by freezing pre-trained model weights or mixing layer-wise representations, as in MTO-PINNs for traffic prediction (Wang et al., 2023). Adaptive mixing coefficients $\alpha_{k,m}^{(\ell)}$ decide dynamically how much information to transfer between tasks.

4. Benchmark Applications and Empirical Performance

Physics-informed multi-task joint learning now underpins state-of-the-art models across a range of domains:

Framework	Task Types	Key Physics Constraint(s)	Metrics/Performance
MR-PI-RNN (Taneja et al., 2023)	Motion prediction + parameter ID	ODE muscle dynamics	$R^2$ up to 0.88 (4-scale), <1% param error
MH-PINN (Zou et al., 2023)	Multitask PDE regression, generative UQ	Problem-specific PDEs	2x–100x reduction in $L^2$-error
MGL-TRF (Xu et al., 20 Nov 2025)	Lane-change + trajectory GANs	Car-following/Lane-change ODEs	BE/TE/PE ↓12.9–34.9% vs. seq.
PiGRN (Kumar et al., 29 Aug 2024)	Multijoint kinematics/torque + load pred	Biomechanical rigid-body ODE	Torque RMSE 4–11%, $r$ to 0.98
PI-MJCA-BiGRU (Ma et al., 14 Nov 2025)	Muscle force/activation/coordination	Multi-joint rigid-body Hill models	$R^2$ > 0.8, real-time inference
P-M-GP (Zhang et al., 15 Oct 2025)	Spatiotemporal multi-variable GP	Reaction–diffusion PDEs	60% error reduction vs. M-GP PINN
AW-EL-PINNs (Li et al., 28 Sep 2025)	OC state/costate/control	Euler–Lagrange BVP	$\mathcal{L}_2$ err $<10^{-3}$
MTO-PINN (Wang et al., 2023)	Traffic density/speed, auxiliary tasks	Kinematic wave PDE	MAPE ↓30% vs. vanilla PINN

Across applications, the introduction of multi-task physics-informed objectives yields substantial improvements over single-task or pure data-driven approaches, particularly in regimes with limited labeled data or under noisy/incomplete observation.

5. Regularization, Uncertainty Quantification, and Generative Modeling

Physics-informed multi-task frameworks offer natural platforms for UQ and generative modeling. The MH-PINN introduces a two-stage method: after multi-task physics-constrained fitting, a normalizing flow is trained over the “heads” (solution coefficients) to yield a generative prior, enabling uncertainty-covering point estimates and few-shot transfer for out-of-distribution tasks (Zou et al., 2023).

Bayesian multi-task PINNs leverage reversible MCMC sampling with adaptive weighting to efficiently explore high-dimensional Pareto fronts, yielding consistent coverage (95–99%) and reduced bias under noise and task heterogeneity (Perez et al., 2023). Empirical results indicate that uncertainty weights—whether learned or inferred via posterior gradient statistics—are critical for stable, well-calibrated physics-informed inference.

Regularization is further enacted via parameter-sharing schemes (soft and hard), dynamically-mixed network weights, and pre-training on auxiliary physics-derived data; ablation studies confirm these significantly outperform equal-weighted or pure physics/data loss approaches (Ma et al., 4 Jun 2024, Bahmani et al., 2021).

6. Discussion and Outlook

Physics-informed multi-task joint learning achieves an overview between deep learning flexibility, multi-objective optimization, and domain-specific scientific priors. Successes in musculoskeletal modeling, spatiotemporal system identification, control, and neuroimaging signal decoding highlight the generality of the approach. Several open directions remain:

• Automating loss-weight tuning (e.g., via Bayesian optimization or fully Bayesian task-uncertainty inference).
• Extending curricula (e.g., multi-resolution or transfer learning) to high-dimensional PDE systems.
• Scaling generative modeling (normalizing flows, GANs) to more realistic scientific data.
• Integrating semi/unsupervised or pretext tasks for low-supervision settings.

Limitations persist: sensitivity to wavelet or kernel design in multi-resolution or spatiotemporal GPs, complexity of balancing many heterogeneous tasks, and need for theoretically-grounded regularization for high-noise, high-uncertainty tasks. However, physics-informed multi-task joint learning is now established as a foundational paradigm for scientific machine learning, combining interpretability, robustness, and predictive performance beyond what either physics-free or single-task models can achieve (Taneja et al., 2023, Zou et al., 2023, Xu et al., 20 Nov 2025, Kumar et al., 29 Aug 2024, Zhang et al., 15 Oct 2025, Li et al., 28 Sep 2025, Wang et al., 2023, Perez et al., 2023, Bahmani et al., 2021).