Multi-Input Operator Network (MIONet)

• Multi-Input Operator Network (MIONet) is a neural architecture that approximates nonlinear operators from multiple input functions using dedicated branch and trunk networks.
• It employs separate branch nets to encode individual function inputs and a trunk net to integrate spatial coordinates via elementwise fusion.
• Empirical results demonstrate MIONet’s effectiveness in PDEs and dynamical systems, achieving low error rates and significant computational speedups.

A Multi-Input Operator Network (MIONet) is a neural architecture designed to learn nonlinear operators that map vectors of functions, or “multi-inputs,” drawn from a product of infinite- or finite-dimensional function spaces (such as Banach or metric spaces), to functions in an output space. By leveraging a modular structure—typically multiple “branch nets” and a single “trunk net”—MIONet generalizes architectures like DeepONet and FNO to problems where the governing mapping depends not just on a single input function but on multiple inputs of disparate type. MIONet and its variants have been employed and extended in operator learning for scientific machine learning (PDE surrogates, dynamical systems, coupled multiphysics), structural response prediction, virtual sensing, model predictive control, and other contexts where operators naturally depend on multiple inputs of different modalities or parameterizations (Jin et al., 2022, Xiao et al., 23 Feb 2024, Ahmed et al., 11 May 2025, Kobayashi et al., 28 Nov 2024).

1. Universal Approximation Theory for Multi-Input Operators

The foundational result underlying MIONet is an extension of the universal approximation theorem to operators defined on products of Banach or metric spaces (Jin et al., 2022, Xiao et al., 23 Feb 2024). Given a continuous operator

$$\mathcal{G}: K_1 \times K_2 \times \ldots \times K_n \to Y,$$

where $K_i$ are compact subsets of Banach or metric spaces $X_i$, MIONet can approximate $\mathcal{G}$ arbitrarily well. This is achieved by constructing “coordinate” projections $\phi_{q_i}$ (typically via Schauder basis expansion or metric discretization) for each input, such that for sufficiently large $q_i$,

$$\sup_{v_i \in K_i} \left\| \mathcal{G}(v_1, \ldots, v_n) - \sum_j g_j^1(\phi_{q_1}^1(v_1))\cdots g_j^n(\phi_{q_n}^n(v_n)) u_j \right\| < \epsilon,$$

with $g_j^k$ continuous coordinate functions (approximated by branch networks) and $u_j$ output basis functions (Jin et al., 2022). The crucial insight is the injective tensor product structure

$$C(K_1 \times \ldots \times K_n, Y) \simeq C(K_1) \otimes \cdots \otimes C(K_n) \otimes Y,$$

enabling the operator to be decomposed into low-rank or full tensor products of finite-dimensional encodings of each input.

2. Network Architecture: Branch and Trunk Nets

The standard MIONet architecture consists of:

• Multiple branch networks: Each branch net independently encodes a discretized version of a single input function $v_i$ (e.g., via $\phi_{q_i}(v_i)$) to yield a latent vector. Branch nets can be classical feed-forward networks, convolutional networks (e.g., U-Net blocks for high-dimensional spatial data (Diab et al., 2023)), LSTMs for sequential data (Kong et al., 2023), or incorporate physical priors.
• A single trunk network: The trunk net encodes the coordinates $y$ at which output is to be evaluated, often producing a high-dimensional set of features.
• Fusion/mixing: The latent branch representations are merged by elementwise multiplication (Hadamard product, low-rank realization) or, more generally, a learned tensor contraction, and this merged representation is projected onto the trunk output by a dot product and summed to yield the final output.

Formally, the typical output is written as:

$$\mathcal{G}(v_1, ..., v_n)(y) = \mathcal{S}\left( b_1(\phi_{q_1}(v_1)) \circledast b_2(\phi_{q_2}(v_2)) \circledast \cdots \circledast f(y) \right) + b,$$

where $\circledast$ is the Hadamard product, $b_i$ are the branch nets, $f$ is the trunk, and $\mathcal{S}(\cdot)$ represents summing over the feature index (Jin et al., 2022, Jiang et al., 2023).

Advancements include multiple trunk networks to explicitly decouple space and time for dynamic responses (Ahmed et al., 11 May 2025), recurrent branch nets for history or sequence inputs (Kong et al., 2023), and periodic layers or U-Net branches for encoding physical priors.

3. Theoretical Analysis and Model Error

Approximation errors in MIONet stem from two principal sources:

• Projection error: Replacing an infinite-dimensional input by its finite coordinate representation via $\phi_{q_i}$ introduces an error that decays as $q_i$ increases.
• Operator error: Approximating the operator’s tensor contraction or function combination (by finite-width networks) also limits accuracy.

The combined error bound is of the form

$$\text{Error} \leq \epsilon + M\sum_{i=1}^n L_i^\epsilon(q_i),$$

with $L_i^\epsilon(q_i) \to 0$ as $q_i \to \infty$ (Jin et al., 2022). For PDE applications with variable domains, the theory has been extended to metric spaces with projection assumptions guaranteeing dense recovery of domain representations (Xiao et al., 23 Feb 2024).

The modular structure of MIONet also allows for explicit incorporation of prior knowledge:

• Linearity: If the operator is linear in $v_j$, the corresponding branch can be implemented as a linear layer.
• Periodicity: For outputs defined over periodic spatial domains, the trunk is augmented with periodic functions (cos/sin).

Adopting task-aligned architectural modifications (e.g., physics-mimetic combination in VarMiON (Patel et al., 2022), or AP loss for high-field kinetic regimes (Zhang et al., 20 Jul 2024)) further improves robustness and physical consistency.

4. Application Domains and Empirical Performance

MIONet and its close analogs have been successfully applied to a range of scientific and engineering tasks:

Application	Input Types	Output	Performance Notes
ODE/PDE solution operators (Jin et al., 2022)	Coefficient fields, initial conditions	Solution function	<2% L² error on nonlinear ODEs/PDEs; MIONet outperforms DeepONet
PDEs on varying domains (Xiao et al., 23 Feb 2024)	Domain geometry, coefficients, BCs	PDE solution	3% rel. L² error across polygons, nonconvex domains, meshless prediction
Structural dynamics (Ahmed et al., 11 May 2025)	Load type, velocity, mesh, time	Displacement/rotation field	<5% error, 100× faster than FEM, continuous-in-time prediction
Virtual sensors (Kobayashi et al., 28 Nov 2024)	Power profile, inlet T, v	Temp/velocity/turbulence	<1% rel. error, $>10^5$ speedup over CFD, resilient to dataset shift
Kinetic equations, AP learning (Zhang et al., 20 Jul 2024)	Initial condition $f_0$, background	$(f, \rho, \phi)$	$<0.2\%$ error in density, no explicit equilibrium needed
Multiphysics and coupled problems (Kobayashi et al., 4 Jul 2025)	Source, boundary, material inputs	Fields (e.g., T, stress)	Multi-branch (MIONet) excels for decoupled, single-branch for tight coupling

In coupled multiphysics, empirical evidence shows MIONet-style (multi-branch) networks offer superior performance for weakly coupled or decoupled single-physics problems, whereas single-branch/shared-parameter designs are favored for strong nonlinear coupling (Kobayashi et al., 4 Jul 2025). This suggests the architecture should reflect the physical coupling regime for optimal results.

5. Extensions: Dynamic, Real-Time, and Multi-Operator Variants

The MIONet framework has been generalized across multiple axes:

• Time-Dependent Systems: Sequential or LSTM-based branch nets (B-LSTM-MIONet) efficiently handle variable-length temporal data and real-time inference, outperforming feedforward architectures for non-uniform, history-dependent problems (Kong et al., 2023). MS-DeepONet provides single-shot multi-step prediction for MPC (Jong et al., 23 May 2025).
• Physics-Informed and Hybrid Iterative Methods: By integrating physics-informed loss (consistency with structural dynamic equilibrium, mass conservation, or other laws), MIONet surrogates achieve accuracy and stability without re-solving PDEs during training (Ahmed et al., 11 May 2025, Zhang et al., 20 Jul 2024, Hu et al., 11 Feb 2024). In hybrid iterative methods, periodic MIONet corrections eliminate low-frequency error modes in iterative solvers, providing meshless acceleration (Hu et al., 11 Feb 2024).
• Meshless, Domain-Agnostic Solvers: Learning operators on variable domains by mapping geometric information into a canonical (e.g., unit ball) domain via metric-space discretizations enables meshless training and evaluation, facilitating general solvers for parameterized PDEs (Xiao et al., 23 Feb 2024).
• Multi-Operator/Meta/Fine-Tuning Paradigms: MODNO adopts distributed training across operators, centralizing branch encoding and decentralizing trunk bases, which yields efficient adaptation to multiple PDE families with reduced data and parameter cost (Zhang, 3 Apr 2024). LeMON demonstrates further that pretraining on diverse operator families and conditioning models on PDE encodings enables effective zero-shot or few-shot generalization to unseen operators, especially when combined with low-rank adaptation and PDE-agnostic meta-learning (Sun et al., 28 Aug 2024).

6. Practical Implementation and Computational Considerations

MIONet models are modular and easily adapted to input heterogeneity, prior knowledge, and problem-specific constraints:

• Computational Efficiency: Compared to full deep ensembles or recurrent architectures, MIONet-type surrogates can deliver orders-of-magnitude speedup ($10^2$–$10^5\times$) vs. high-fidelity solvers, with only marginal increases in parameter count and inference cost when leveraging shared weights and efficient mixing (Kobayashi et al., 28 Nov 2024, Jiang et al., 2023, Ahmed et al., 11 May 2025).
• Training Strategies: Consider automatic hyperparameter tuning (as in MS-DeepONet (Jong et al., 23 May 2025)), meshless or semi-supervised data generation (for variable domains or operator families), and embedding physical structure (physics-informed loss, periodic trunk nets, etc.) to stabilize and accelerate convergence while enhancing generalization.
• Error Decomposition and Robustness: The error in MIONet surrogates can be systematically decomposed (projection, approximation, training, quadrature, and covering error). This enables theoretical and empirical guarantees on convergence, robustness to sensor/mesh layout changes, and adaptability to dataset shift (Patel et al., 2022, Xiao et al., 23 Feb 2024, Kobayashi et al., 28 Nov 2024).
• Design Principles: Choose multi-branch/multi-trunk architectures for decoupled or heterogeneously parametrized problems; use shared-parameter encodings or single-branch designs for tightly coupled multiphysics scenarios (Kobayashi et al., 4 Jul 2025). Incorporate priors (e.g., linearity, periodicity, conservation laws) directly in the corresponding branches/trunks or via transformation layers.

7. Outlook and Ongoing Developments

MIONet and its related operator learning frameworks are driving foundational advances in scientific machine learning, surrogate modeling, and real-time monitoring. Ongoing trends include:

• Data efficiency and meta-learning: Leveraging foundation model pretraining and meta-learning yields rapid adaptation to new PDEs and operator classes with minimal data (Sun et al., 28 Aug 2024).
• Hybrid and physics-mimetic models: Integration with mechanistic solvers or variational principles robustly addresses the accuracy/stability challenges posed by complex physical systems (Patel et al., 2022, Ahmed et al., 11 May 2025, Hu et al., 11 Feb 2024).
• Real-world applications: From digital twins for structural health monitoring (Ahmed et al., 11 May 2025) to virtual sensors in inaccessible environments (Kobayashi et al., 28 Nov 2024), MIONet-based surrogates are deployed in environments demanding both flexibility and rigor.

A plausible implication is that future MIONet development will focus on coupling-aware architecture selection, scalable transfer learning across operator families, and adaptive hybridization with classical solvers, with special attention to meshless, real-time, and multi-modal inference (Kobayashi et al., 4 Jul 2025, Zhang, 3 Apr 2024, Sun et al., 28 Aug 2024).