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mapping from the parameters describing the geometry to the velocity and pressure felds.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 700000, "bbox": [[0.735, 0.429, 0.765, 0.444], [0.235, 0.443, 0.765, 0.458], [0.235, 0.457, 0.525, 0.472]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 17, "edu_l1_label": "IOS"}, {"txt": "While the DeepONet is solely datadriven, the PC-DeepONet imposes the divergence constraint from the continuity equation onto the network.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 700000, "bbox": [[0.525, 0.457, 0.765, 0.472], [0.235, 0.471, 0.765, 0.486], [0.235, 0.485, 0.408, 0.5]]}]}, "tags": ["text"], "label": "abstract", "web_segment_id": 0, "global_sentence_id": 18, "edu_l1_label": "IOS"}, {"txt": "The PC-DeepONet demonstrates higher accuracy than the data-driven baseline, especially when 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pressure, µ is the dynamic viscosity and fb the resultantof the body forces (such as gravity and centrifugal forces) per unit volume acting on the continuum.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1300000, "bbox": [[0.176, 0.15, 0.824, 0.165], [0.176, 0.164, 0.824, 0.179]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 60, "edu_l1_label": "IOS"}, {"txt": "PINNs have been used as surrogate models for CFD as they are able to predict fows for cases where little/no data is available Raissi et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1300000, "bbox": [[0.176, 0.178, 0.824, 0.193], [0.176, 0.192, 0.431, 0.207]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 61, "edu_l1_label": "IOS"}, {"txt": "(2019); Arzani et al.", "language": "english", "position": {"pdf_position": 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[{"page_number": 0, "shape": [1650, 1275], "position_id": 1400000, "bbox": [[0.711, 0.282, 0.772, 0.298]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 70, "edu_l1_label": "IOS"}, {"txt": "(2019), is explored to create a surrogate for predicting fow over curved backward-facing steps (BFS) with different parameterised curves.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1400000, "bbox": [[0.772, 0.282, 0.824, 0.298], [0.176, 0.296, 0.824, 0.311], [0.176, 0.31, 0.378, 0.325]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 5, "global_sentence_id": 71, "edu_l1_label": "IOS"}, {"txt": "Embedding fundamental principles, like the continuity equation, directly into the structure of neural networks can enhance the effcacy of cutting-edge neural operator models by narrowing down the potential solution space for our models Richter-Powell et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1500000, "bbox": [[0.176, 0.331, 0.824, 0.347], [0.176, 0.345, 0.824, 0.36], [0.176, 0.359, 0.545, 0.374]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 72, "edu_l1_label": "IOS"}, {"txt": "(2022).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1500000, "bbox": [[0.545, 0.359, 0.595, 0.374]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 73, "edu_l1_label": "IOS"}, {"txt": "We propose a Physics-Constrained DeepONet (PC-DeepONet), an architecture that exactly satisfes the continuity equation (Eq 1).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1500000, "bbox": [[0.595, 0.359, 0.824, 0.374], [0.176, 0.373, 0.797, 0.388]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 74, "edu_l1_label": "IOS"}, {"txt": "We show, through numerical experiments, that the proposed architecture increases the accuracy for the surrogate model, converging in only a few iterations with sparse training data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1500000, "bbox": [[0.797, 0.373, 0.824, 0.388], [0.176, 0.387, 0.824, 0.402], [0.176, 0.401, 0.683, 0.416]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 75, "edu_l1_label": "IOS"}, {"txt": "We investigate the use of PC-DeepOnets through the lens of the backward-facing step (BFS).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.176, 0.422, 0.793, 0.437]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 76, "edu_l1_label": "IOS"}, {"txt": "The BFS is a benchmark model that demonstrates important features such as fow separation, vortex evolution and re-attachment Chen et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.793, 0.422, 0.824, 0.437], [0.176, 0.436, 0.824, 0.451], [0.176, 0.45, 0.437, 0.465]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 77, "edu_l1_label": "IOS"}, {"txt": "(2018).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.437, 0.45, 0.488, 0.465]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 78, "edu_l1_label": "IOS"}, {"txt": "The BFS has been studied by multiple researchers, especially in the felds of aerospace Armaly et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.488, 0.45, 0.824, 0.465], [0.176, 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"language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.28, 0.491, 0.824, 0.506], [0.176, 0.505, 0.824, 0.52], [0.176, 0.519, 0.351, 0.534]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 85, "edu_l1_label": "IOS"}, {"txt": "(2012), making it suitable for learning the operator mapping between the shape of the curve and the resulting fow.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1600000, "bbox": [[0.351, 0.519, 0.824, 0.534], [0.176, 0.533, 0.444, 0.548]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 86, "edu_l1_label": "IOS"}, {"txt": "Furthermore, curved surfaces are common in engineering applications, for example in aerodynamic bodies, blades, curved ducts and pipes, cylinders, and diffusers Bentaleb et al.", "language": "english", "position": {"pdf_position": 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and the fow feld evaluated on the output grid.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1800000, "bbox": [[0.176, 0.625, 0.824, 0.64], [0.176, 0.638, 0.718, 0.654]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 90, "edu_l1_label": "IOS"}, {"txt": "The DeepONet was chosen as it works well for learning functions between parameterised inputs and their corresponding outputs.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 1800000, "bbox": [[0.718, 0.638, 0.824, 0.654], [0.176, 0.652, 0.824, 0.667], [0.176, 0.666, 0.293, 0.681]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 91, "edu_l1_label": "IOS"}, {"txt": "An illustration of the geometry of choice can be found in Appendix A.1(Figure 2).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 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"content", "web_segment_id": 6, "global_sentence_id": 104, "edu_l1_label": "IOS"}, {"txt": "FreeFEM++ uses the fnite element method to solve the incompressible Navier-Stokes equations", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2100000, "bbox": [[0.176, 0.91, 0.824, 0.925]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 6, "global_sentence_id": 105, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2200000, "bbox": [[0.494, 0.95, 0.507, 0.959]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 106, "edu_l1_label": "EDU_O"}, {"txt": "header", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2200000, "bbox": [[0.176, 0.034, 0.693, 0.047]]}]}, "tags": ["header"], "label": "O", "web_segment_id": -1, 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For solution control, the incompressible fow solver combined a number of numerical techniques, such as an implicit scheme for time-stepping, fnite element spatial discretization, and iterative solvers for managing the resulting linear systems. To ensure accurate representation of physical phenomena, stabilization techniques such as the SUPG (Streamline Upwind/Petrov-Galerkin) method Brooks & Hughes (1982), which is particularly benefcial for handling convection-dominated fows, was employed. The simulations were run for 2D unstructured meshes, each comprising approximately 50000 elements, across 50 distinct geometries. Uniform boundary conditions were applied to all simulations. The fow was modeled as incompressible, with a fxed Reynolds number of 1000.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2400000, "bbox": [[0.173, 0.253, 0.829, 0.38]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 7, "global_sentence_id": 110, "edu_l1_label": "IOS"}, {"txt": "2.2NETWORK ARCHITECTURE", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2500000, "bbox": [[0.177, 0.394, 0.401, 0.408]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 5, "global_sentence_id": 111, "edu_l1_label": "IOS"}, {"txt": "The baseline DeepONet, as proposed by Lu et al.Lu et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.176, 0.42, 0.573, 0.435]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 112, "edu_l1_label": "IOS"}, {"txt": "(2019) consists of two sub-networks: a branch network and a trunk network as illustrated in Figure 3 in Appendix A.1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.573, 0.42, 0.824, 0.435], [0.176, 0.433, 0.711, 0.449]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 113, "edu_l1_label": "IOS"}, {"txt": "The input to the branch network consists of the x-coordinates of the control point pair for each geometry confguration.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.711, 0.433, 0.824, 0.449], [0.176, 0.447, 0.824, 0.462], [0.176, 0.461, 0.206, 0.476]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 114, "edu_l1_label": "IOS"}, {"txt": "Hence, the size of the input to the branch net is the number of geometries x 2.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.206, 0.461, 0.728, 0.476]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 115, "edu_l1_label": "IOS"}, {"txt": "The trunk net takes as input the output sensors: the spatial coordinates of the output mesh.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.728, 0.461, 0.824, 0.476], [0.176, 0.475, 0.669, 0.49]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 116, "edu_l1_label": "IOS"}, {"txt": "The size of the input to the trunk net is therefore the number of sensors x dimensions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.669, 0.475, 0.824, 0.49], [0.176, 0.489, 0.583, 0.504]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 117, "edu_l1_label": "IOS"}, {"txt": "Three separate DeepONet networks with the same architecture were trained to predict the horizontal velocity (U), vertical velocity (V) and pressure(P) felds.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.583, 0.489, 0.824, 0.504], [0.176, 0.503, 0.824, 0.518], [0.176, 0.516, 0.323, 0.532]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 118, "edu_l1_label": "IOS"}, {"txt": "For the PC-DeepONet, this architecture was extended by incorporating the approach proposed by Richter et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.323, 0.516, 0.824, 0.532], [0.176, 0.53, 0.404, 0.545]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 119, "edu_l1_label": "IOS"}, {"txt": "Richter-Powell et al.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.404, 0.53, 0.539, 0.545]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 120, "edu_l1_label": "IOS"}, {"txt": "(2022) to make the velocity feld divergence free .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2600000, "bbox": [[0.539, 0.53, 0.824, 0.545], [0.176, 0.544, 0.21, 0.559]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 121, "edu_l1_label": "IOS"}, {"txt": "The output of an individual DeepONet can be expressed as follows:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2700000, "bbox": [[0.176, 0.566, 0.617, 0.581]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 122, "edu_l1_label": "IOS"}, {"txt": "GCα(x,y) = 2", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.313, 0.608, 0.409, 0.631]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 15, "global_sentence_id": 123, "edu_l1_label": "IOS"}, {"txt": "Nα", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.401, 0.598, 0.424, 0.615]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 16, "global_sentence_id": 124, "edu_l1_label": "IOS"}, {"txt": "三=1", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.4, 0.627, 0.424, 0.641]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 13, "global_sentence_id": 125, "edu_l1_label": "IOS"}, {"txt": "江4(69;", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.398, 0.61, 0.685, 0.63]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 126, "edu_l1_label": "IOS"}, {"txt": "θα)Φ1(x,y;", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.398, 0.61, 0.685, 0.63]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 127, "edu_l1_label": "IOS"}, {"txt": "θΦ)Vq∈{p,u,u}.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.398, 0.61, 0.685, 0.63]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 14, "global_sentence_id": 128, "edu_l1_label": "IOS"}, {"txt": "(3)", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2800000, "bbox": [[0.805, 0.611, 0.824, 0.626]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 9, "global_sentence_id": 129, "edu_l1_label": "IOS"}, {"txt": "Where ϕ a collection of basis functions that map functions from spatial coordinates to output func-tions, and α are a collection of coeffcients that learn mappings from the parametrized slope controlpoints to the output functions.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2900000, "bbox": [[0.176, 0.648, 0.824, 0.663], [0.176, 0.662, 0.824, 0.677], [0.176, 0.676, 0.372, 0.691]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 130, "edu_l1_label": "IOS"}, {"txt": "Considering the two individual DeepONet output b(x, y) as a vector feld with components bu(x, y) and bv(x, y) corresponding to the velocities u and v, we have:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 2900000, "bbox": [[0.372, 0.676, 0.824, 0.691], [0.176, 0.69, 0.786, 0.705]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 131, "edu_l1_label": "IOS"}, {"txt": "b ( x , y ) = \\left[ \\begin{matrix} { b _ { u } ( x , y ) } \\\\ { b _ { v } ( x , y ) } \\\\ \\end{matrix} \\right] = \\left[ \\begin{matrix} { G ^ { \\xi _ { u } } ( x , y ) } \\\\ { G ^ { \\xi _ { u } } ( x , y ) } \\\\ \\end{matrix} \\right] = \\left[ \\begin{matrix} { \\sum _ { i = 1 } ^ { N _ { \\alpha } } \\alpha _ { i } ^ { u } ( \\xi _ { g } ; \\theta _ { \\alpha } ) \\phi _ { i } ^ { u } ( x , y ; \\theta _ { \\phi } ) } \\\\ { \\sum _ { i = 1 } ^ { N _ { \\alpha } } \\alpha _ { i } ^ { v } ( \\xi _ { g } ; \\theta _ { \\alpha } ) \\phi _ { i } ^ { v } ( x , y ; \\theta _ { \\phi } ) } \\\\ \\end{matrix} \\right] .", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3000000, "bbox": [[0.25, 0.718, 0.747, 0.757]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 18, "global_sentence_id": 132, "edu_l1_label": "IOS"}, {"txt": "(4)", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3000000, "bbox": [[0.805, 0.732, 0.824, 0.747]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 10, "global_sentence_id": 133, "edu_l1_label": "IOS"}, {"txt": "With this vector feld, a skew-symmetric matrix A is constructed from the Jacobian of b and its transpose:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3100000, "bbox": [[0.176, 0.766, 0.824, 0.781], [0.176, 0.78, 0.242, 0.795]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 134, "edu_l1_label": "IOS"}, {"txt": "A = J _ { b } - J _ { b } ^ { \\top } = \\left[ \\begin{matrix} { 0 } & { \\frac { \\partial b _ { u } } { \\partial y } - \\frac { \\partial b _ { v } } { \\partial x } } \\\\ { \\frac { \\partial b _ { v } } { \\partial x } - \\frac { \\partial b _ { u } } { \\partial y } } & { 0 } \\\\ \\end{matrix} \\right] ,", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3200000, "bbox": [[0.345, 0.809, 0.648, 0.853]]}]}, "tags": ["equation"], "label": "formula", "web_segment_id": 17, "global_sentence_id": 135, "edu_l1_label": "IOS"}, {"txt": "(5)", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3200000, "bbox": [[0.805, 0.824, 0.824, 0.84]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 11, "global_sentence_id": 136, "edu_l1_label": "IOS"}, {"txt": "where Jb is the Jacobian matrix of b with regards to the trunk input.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3300000, "bbox": [[0.176, 0.861, 0.644, 0.876]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 137, "edu_l1_label": "IOS"}, {"txt": "The fnal divergence-free velocity feld v is represented by this divergence:", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3300000, "bbox": [[0.644, 0.861, 0.824, 0.876], [0.176, 0.875, 0.495, 0.89]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 138, "edu_l1_label": "IOS"}, {"txt": "v = div(A).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3400000, "bbox": [[0.457, 0.912, 0.54, 0.925]]}]}, "tags": ["text"], "label": "formula", "web_segment_id": 12, "global_sentence_id": 139, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.805, 0.913, 0.826, 0.924]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 140, "edu_l1_label": "EDU_O"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.494, 0.949, 0.507, 0.961]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 141, "edu_l1_label": "EDU_O"}, {"txt": "header", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3500000, "bbox": [[0.176, 0.033, 0.694, 0.047]]}]}, "tags": ["header"], "label": "O", "web_segment_id": -1, "global_sentence_id": 142, "edu_l1_label": "EDU_O"}, {"txt": "By construction, the divergence of A is zero due to the properties of the skew-symmetric matrix, ensuring that the predicted velocity feld is divergence-free, which aligns with the continuity condition.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3600000, "bbox": [[0.176, 0.104, 0.824, 0.119], [0.176, 0.118, 0.824, 0.133], [0.176, 0.132, 0.206, 0.147]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 143, "edu_l1_label": "IOS"}, {"txt": "2.3TRAINING AND TESTING", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3700000, "bbox": [[0.177, 0.168, 0.385, 0.182]]}]}, "tags": ["title"], "label": "title2", "web_segment_id": 6, "global_sentence_id": 144, "edu_l1_label": "IOS"}, {"txt": "Our model’s implementation and optimization were conducted using the PyTorch framework.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.176, 0.195, 0.779, 0.21]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 145, "edu_l1_label": "IOS"}, {"txt": "Train-ing was executed on a dataset comprising 50 distinct geometries.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.779, 0.195, 0.824, 0.21], [0.176, 0.209, 0.598, 0.224]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 146, "edu_l1_label": "IOS"}, {"txt": "For each geometry, we performed a mesh mapping to a target geometry to obtain a unique mesh using bi-cubic interpolation.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.598, 0.209, 0.824, 0.224], [0.176, 0.223, 0.782, 0.238]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 147, "edu_l1_label": "IOS"}, {"txt": "1000 output sensors were then randomly positioned across the mesh in points away from the slope to train the model.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.782, 0.223, 0.824, 0.238], [0.176, 0.237, 0.824, 0.252], [0.176, 0.25, 0.246, 0.266]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 148, "edu_l1_label": "IOS"}, {"txt": "Since the mapping is done from function to function rather than from vector to vector, the model does not have to be trained on all the points in the domain and can learn the representation more effciently with fewer sampled points.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.246, 0.25, 0.824, 0.266], [0.176, 0.264, 0.824, 0.279], [0.176, 0.278, 0.458, 0.293]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 149, "edu_l1_label": "IOS"}, {"txt": "The DeepONet was then able to extrapolate for individual geometries, even in unseen domains.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.458, 0.278, 0.824, 0.293], [0.176, 0.292, 0.441, 0.307]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 150, "edu_l1_label": "IOS"}, {"txt": "We illustrate this resolution-free capability of DeepONets by infering on a mesh with 50000 points, as shown in Figures 4 and 5in Appendix A.1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3800000, "bbox": [[0.441, 0.292, 0.824, 0.307], [0.176, 0.306, 0.751, 0.321]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 151, "edu_l1_label": "IOS"}, {"txt": "The optimization of the network parameters was performed using the Limited-memory BroydenFletcher-Goldfarb-Shanno (L-BFGS) algorithm. This particular optimizer was chosen to address vanishing gradient issues that arise when training DeepONets with hyperbolic tangents (tanh). Only 50 iterations were needed to achieve convergence for both architectures. The hyperparameters used for both DeepONet models are displayed in Table 1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 3900000, "bbox": [[0.173, 0.327, 0.827, 0.398]]}]}, "tags": ["table_caption"], "label": "content", "web_segment_id": 10, "global_sentence_id": 152, "edu_l1_label": "IOS"}, {"txt": "figure", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4000000, "bbox": [[0.322, 0.41, 0.675, 0.504]]}]}, "tags": ["figure"], "label": "figure", "web_segment_id": 7, "global_sentence_id": 153, "edu_l1_label": "EDU_O"}, {"txt": "Table 1: Hyperparameters of the Optimized DeepONet", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4100000, "bbox": [[0.32, 0.511, 0.681, 0.527]]}]}, "tags": ["table_caption"], "label": "table_title", "web_segment_id": 9, "global_sentence_id": 154, "edu_l1_label": "EDU_O"}, {"txt": "To evaluate the model’s performance, we used the Mean Squared Error (MSE) loss function andassessed the relative L2 error between the predicted and actual sensor data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4200000, "bbox": [[0.175, 0.544, 0.826, 0.574]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 11, "global_sentence_id": 155, "edu_l1_label": "IOS"}, {"txt": "3RESULTS AND DISCUSSION", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4300000, "bbox": [[0.177, 0.595, 0.433, 0.613]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 5, "global_sentence_id": 156, "edu_l1_label": "BOS"}, {"txt": "We report the performance of the vanilla DeepONet and the PC-DeepONet models in Table 2.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.176, 0.63, 0.792, 0.645]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 157, "edu_l1_label": "IOS"}, {"txt": "The PC-DeepONet, shows superior performance in both training and validation phases.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4400000, "bbox": [[0.792, 0.63, 0.824, 0.645], [0.176, 0.644, 0.716, 0.659]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 4, "global_sentence_id": 158, "edu_l1_label": "IOS"}, {"txt": "Both models were tested on unseen domains and geometries.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4500000, "bbox": [[0.176, 0.665, 0.577, 0.68]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 159, "edu_l1_label": "IOS"}, {"txt": "The outputs from the trained models on a case from the validation set are displayed in Figure 4 and 5 in Appendix A.1.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4500000, "bbox": [[0.577, 0.665, 0.824, 0.68], [0.176, 0.679, 0.718, 0.694]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 2, "global_sentence_id": 160, "edu_l1_label": "IOS"}, {"txt": "Table 2: Performance comparison of DeepONet and PC-DeepONet.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4600000, "bbox": [[0.278, 0.722, 0.723, 0.735]]}]}, "tags": ["table_caption"], "label": "table_title", "web_segment_id": 8, "global_sentence_id": 161, "edu_l1_label": "EDU_O"}, {"txt": "table", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4700000, "bbox": [[0.225, 0.735, 0.769, 0.783]]}]}, "tags": ["table"], "label": "table", "web_segment_id": 1, "global_sentence_id": 162, "edu_l1_label": "EDU_O"}, {"txt": "Both architectures achieved convergence within a limited number of iterations. The PC-DeepONet, with its enforced continuity condition, demonstrates superior accuracy with sparse data. We note that these results were obtained under conditions of smooth, laminar fow. We anticipate that for turbulent fows, the advantage of the PC-DeepONet will become more pronounced. Despite the resolution free-nature of neural operators, analysis of the Figures 4 and 5 reveals that errors cluster near boundary layers, likely due to under-sampling in the regions close to the slope, where points cannot be sampled in a common mesh. Future investigations will explore masking approaches to incorporate this information in the training and incorporating hard boundary conditions in the architecture.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4800000, "bbox": [[0.173, 0.8, 0.827, 0.925]]}]}, "tags": ["figure_caption"], "label": "content", "web_segment_id": 12, "global_sentence_id": 163, "edu_l1_label": "IOS"}, {"txt": "footer", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4900000, "bbox": [[0.496, 0.95, 0.506, 0.959]]}]}, "tags": ["footer"], "label": "O", "web_segment_id": -1, "global_sentence_id": 164, "edu_l1_label": "EDU_O"}, {"txt": "header", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 4900000, "bbox": [[0.176, 0.033, 0.693, 0.047]]}]}, "tags": ["header"], "label": "O", "web_segment_id": -1, "global_sentence_id": 165, "edu_l1_label": "EDU_O"}, {"txt": "4CONCLUSION AND FUTURE WORK", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5000000, "bbox": [[0.177, 0.102, 0.494, 0.119]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 12, "global_sentence_id": 166, "edu_l1_label": "BOS"}, {"txt": "In this study, we introduce PC-DeepONets, an architecture that incorporates knowledge of the physics of fuid fow into the DeepONet to improve on data-driven learning.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5100000, "bbox": [[0.176, 0.134, 0.824, 0.149], [0.176, 0.148, 0.697, 0.163]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 167, "edu_l1_label": "IOS"}, {"txt": "By enforcing the divergence constraint imposed by the continuity equation in the DeepONet framework, we observe signifcant improvements in accuracy for surrogate modeling with sparse data.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5100000, "bbox": [[0.697, 0.148, 0.824, 0.163], [0.176, 0.162, 0.824, 0.177], [0.176, 0.176, 0.678, 0.191]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 168, "edu_l1_label": "IOS"}, {"txt": "Using PC-DeepONet, we successfully learned operational mappings between parameterized geometric confgurations and fow feld variables (U, V, and P).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5100000, "bbox": [[0.678, 0.176, 0.824, 0.191], [0.176, 0.19, 0.824, 0.205], [0.176, 0.203, 0.394, 0.219]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 0, "global_sentence_id": 169, "edu_l1_label": "IOS"}, {"txt": "Moving forward, integrating additional impositions by constraining boundaries and integrating additional information from the physics could potentially improve the performance of PC-DeepONets in capturing complex fow phenomena like turbulence for higher Reynolds numbers.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.176, 0.225, 0.824, 0.24], [0.176, 0.239, 0.824, 0.254], [0.176, 0.252, 0.727, 0.268]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 170, "edu_l1_label": "IOS"}, {"txt": "Moreover, we aim to explore effcient methods for implementing derivatives such as the method proposed in Jnini & Vella (2024).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5200000, "bbox": [[0.727, 0.252, 0.824, 0.268], [0.176, 0.266, 0.824, 0.281], [0.176, 0.28, 0.278, 0.295]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 1, "global_sentence_id": 171, "edu_l1_label": "IOS"}, {"txt": "ACKNOWLEDGEMENTS", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5300000, "bbox": [[0.177, 0.314, 0.368, 0.332]]}]}, "tags": ["title"], "label": "title1", "web_segment_id": 14, "global_sentence_id": 172, "edu_l1_label": "EDU_O"}, {"txt": "We would like to thank the organisers of the 2023 Physics-Informed Neural Networks and Applications PhD Summer School at KTH for making this collaboration possible.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.176, 0.347, 0.824, 0.362], [0.176, 0.36, 0.718, 0.376]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 173, "edu_l1_label": "EDU_O"}, {"txt": "We would also like to thank Dr Khemraj Shukla (Assistant Professor of Applied Mathematics (Research), Brown University) for his advice and support during this project.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5400000, "bbox": [[0.718, 0.36, 0.824, 0.376], [0.176, 0.374, 0.824, 0.389], [0.176, 0.388, 0.551, 0.403]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 3, "global_sentence_id": 174, "edu_l1_label": "EDU_O"}, {"txt": "AJ would like to acknowledge Leonardo S.p.A. (Italy).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5500000, "bbox": [[0.176, 0.423, 0.535, 0.439]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 175, "edu_l1_label": "EDU_O"}, {"txt": "HG would like to acknowledge the ART-AI CDT (Bath, UK), funded by the UKRI.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5500000, "bbox": [[0.535, 0.423, 0.824, 0.439], [0.176, 0.437, 0.432, 0.452]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 176, "edu_l1_label": "EDU_O"}, {"txt": "SD would like to acknowledge NSERC (Canada).", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5500000, "bbox": [[0.432, 0.437, 0.76, 0.452]]}]}, "tags": ["text"], "label": "content", "web_segment_id": 9, "global_sentence_id": 177, "edu_l1_label": "EDU_O"}, {"txt": "REFERENCES", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5600000, "bbox": [[0.177, 0.471, 0.286, 0.489]]}]}, "tags": ["title"], "label": "reference", "web_segment_id": 15, "global_sentence_id": 178, "edu_l1_label": "EDU_O"}, {"txt": "Bassem F Armaly, F Durst, JCF Pereira, and B Scho¨nung.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5700000, "bbox": [[0.176, 0.496, 0.547, 0.511]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 8, "global_sentence_id": 179, "edu_l1_label": "EDU_O"}, {"txt": "Experimental and theoretical investigationof backward-facing step fow.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5700000, "bbox": [[0.547, 0.496, 0.824, 0.511], [0.193, 0.51, 0.385, 0.525]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 8, "global_sentence_id": 180, "edu_l1_label": "EDU_O"}, {"txt": "Journal of fuid Mechanics, 127:473–496, 1983.", "language": "english", "position": {"pdf_position": [{"page_number": 0, "shape": [1650, 1275], "position_id": 5700000, "bbox": [[0.385, 0.51, 0.704, 0.525]]}]}, "tags": ["text"], "label": "reference", "web_segment_id": 8, "global_sentence_id": 181, "edu_l1_label": "EDU_O"}, {"txt": "Amirhossein Arzani, Jian-Xun Wang, and Roshan M. 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# PHYSICS-CONSTRAINED DEEPONET FOR SURROGATE CFD MODELS:A CURVED BACKWARD-FACING STEP CASE
## ABSTRACT
## 1INTRODUCTION
## 2METHOD
### 2.1DATA GENERATION
### 2.2NETWORK ARCHITECTURE
### 2.3TRAINING AND TESTING
## 3RESULTS AND DISCUSSION
## 4CONCLUSION AND FUTURE WORK
